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Last Update : 28 April 2013

Geometrikon is a collection of Topics in Geometry and geometric subjects from other fields. It started as a list of examples from Geometry, Euclidean, Affine, and Projective, and evolved for some time to contain proofs and discussions on various topics. The subjects listed below are continuously reviewed and updated with the intention to make the exposition as complete as possible and give enough references to related subjects and bibliography.
From the several hundreds of topics listed below some are rudimentary and intentional for future discussions, some are informal, some are included for aesthetic reasons (like those containing pictures of fractals) and others are detailed expositions with proofs.
The more elaborate topics have already updates and include (i) an html-page and (ii) a EucliDraw document (euc) zipped. The last type of document is created with the Dynamic Geometry application EucliDraw developed at the Geometry Division, Department of Mathematics, University of Crete. Using this Dynamic Geometry environment one can manipulate the figures with the mouse and enjoy the observation of their continuous changes. This needs the last version of the EucliDraw dynamic Geometry application which can be freely downloaded.
The topics below are listed alphabetically using some relevant for the content word of the title.

To suggest a problem or ask a question, please fill free to write to the address: pamfiloXYmathZuocZgr (replacing the letters {X, Y, Z} with{ s, @, . } correspondingly).

To search quickly Geometrikon for an item of your interest, enter the search word into the field (top of the page) and press the button next to it.

 

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

    A

  1. AbridgedNotation      created : 27/08/2009
    The abridged notation used for calculations with projective coordinates in projective geometry.
  2. A4      created : 27/10/2007, updated : 31/01/2008  euc 
    The alternating group of permutations of four elements and its multiplication table.
  3. Adams Circle
    The circle of Adams, concentric to the incircle of a triangle.
  4. Affine Reflexion      created : 06/02/2011
    The concept of Affine Reflexion, definition and first properties.
  5. Affine Reflexion Basics      created : 06/02/2011
    Elementary properties of affine reflexions.
  6. Affinities     created : 27/10/2007, updated : 27/08/2009  euc 
    Transformations preserving lines and their properties.
  7. Affinities, their fixed points      created : 06/02/2011
    A study of the fixed points of Affinities and a classification of Affinities according to their fixed points.
  8. Affinity Generated Parabola     created : 27/08/2009  euc 
    Parabolas constructed out from a given affinity.
  9. Agnesi-curve
    Illustration of the curve "Witch of Agnesi" and description of its generation.
  10. Algebraic_Locus
    A locus generated by some algebraic operations on (complex) points.
  11. All Conics Circumscribed      created : 27/10/2007, updated : 27/08/2009  euc 
    Determination of the family of all conics circumscribing a triangle and tangent to a line.
  12. All Conics Circumscribed (2)
    As before but now the line separating the three vertices of the triangle.
  13. Conics circumscribing a triangle
    A study of the conics circumscribed about a triangle.
  14. All Parabolas Circumscribed       created : 22/06/2006, updated : 31/01/2008, 27/08/09   euc 
    Determination of the family of all parabolas circumscribing a triangle. Some remarks on the two parabolas through four points.
  15. Lines through a fixed point
    A property of lines defined through ratios on the sides of an angle.
  16. Lines defined through ratios
    Defining a line through certain length-ratios on the sides of an angle.
  17. AnalyticVersusSynthetic      15/12/2009, 5/2/10  euc     
    The merits of analytic and synthetic methods of proof in geometry. Study on the ground of an example (locus).
  18. Angle problem      5/01/2010  euc    
    A gem in determining the angles of a simple figure constructed in an isosceles triangle.
  19. Angles on sides of a triangle
    Constructing equal angles on the sides of a triangle. A geometric locus and remarks on Brocard's points.
  20. Angle Multiples
    A tutorial on the various ways to measure angles and their applications.
  21. Anticomplementary and Circumparabola      created : 12/02/2008, updated : 27/08/09   euc  
    A very simple relation of the anticomplementary triangle with a circumparabola of the triangle.
  22. Antiparallel Hyperbola     created : 31/12/2007, updated : 14/02/2008    euc
    A hyperbola connected to the antiparallels to a side of a triangle.
  23. Antiparallels     created : 31/12/2007, updated : 15/12/2009    euc     
    Antiparallels to sides of a triangle, related to the symmedians of a triangle.
  24. Apollonian Circles
    Apollonian circles of a triangle and the bundle they generate (Isodynamic points).
  25. Apollonian Bundle     created : 31/12/2007, updated : 15/12/2009    euc     
    The hyperbolic circle bundle of all Apollonian circles relative to a line segment.
  26. Apollonian Circle Property
    A property of the Apollonian circles of a triangle.
  27. Apollonian pedal property     created : 31/12/2007, updated : 14/02/2008    euc
    Inverting on Apollonian circles preserves the similarity type of a pedal triangle.
  28. Apollonius circles     created : 31/12/2007, updated : 14/02/2008, 06/02/2011    euc
    That the pedals of points of Apollonian circles are isosceli.
  29. Apollonius Problem
    The problem of finding all circles simultaneous tangent to three given circles.
  30. Apollonius Problem-2
    Additional cases out of the multitude, handled directly by the tools of EucliDraw.
  31. Arbelos
    Shoemaker knife and some properties related.
  32. Arc-polygon inscribed in a triangle
    Study of an arc-hexagon inscribed in a triangle, related to the Gergonne point of the triangle.
  33. Arc-hexagon inscribed in a polygon
    Study of an arc-hexagon inscribed in another circumscriptible hexagon.
  34. Arc-hexagon inscribed in a polygon (2)
    Continuation of the previous study, with properties related to the diagonals of the polygons.
  35. Arc-pentagon inscribed in a polygon
    Study of an arc-pentagon inscribed in another circumscriptible pentagon.
  36. Arc-polygon (path) in a quadrangle
    A picture of an arc-polygon inscribed in a general quadrangle.
  37. Oscillation of a point on an arc
    How to construct with EucliDraw an oscillating point on a circular arc > 180 degrees.
  38. Archimedes' Circles
    Two equal in radii circles tangent to the circles of an arbelos.
  39. Area in Barycentric coordinates     created : 31/12/2007, updated : 15/12/2009    euc     
    The area of a triangle expressed in the barycentric coordinates of its vertices.
  40. Area of pedal triangle    created : 31/12/2007, updated : 14/02/2008    euc
    The area in barycentric coordinates of the pedal triangle, formed by the projections of a point on the sides of a triangle.
  41. Area through determinant     created : 31/12/2007, updated : 15/12/2009    euc     
    The expression of the area of a triangle/polygon in terms of the determinant of the coordinates of its vertices.
  42. Area Quotient
    Comparing the areas of a parallelogram and of the quadrangle, which results by reflecting a point on the sides of the parallelogram.
  43. AreasRatio
    Comparing the areas of a triangle and an inscribed to it other triangle.
  44. Artobolevsky's ellipsograph
    A simple link-gear mechanism to draw ellipses and their tangents.
  45. Artzt parabolas of first kind      created : 02/02/2007, updated : 10/10/2008   euc
    Three parabolas associated to a triangle and their relation to the second Brocard triangle.
  46. The three Artzt parabolas of a triangle      created : 31/01/2008, updated : 10/10/2008   euc
    Three parabolas associated to a triangle and their relation to the second Brocard triangle.
  47. Artzt parabola generation III
    The simplest case of an Artzt parabola and its transfer to the general case through affinities.
  48. Artzt parabola generation
    Generating an Artzt parabola as envelope of diagonals of parallelograms (projective coordinates proof).
  49. Artzt parabola generation II
    Generating an Artzt parabola as envelope of diagonals of parallelograms (synthetic proof).
  50. Artzt parabolas of second kind
    Three more parabolas associated to a triangle and their relation to the second Brocard triangle.
  51. Artzt parabola of an isosceles triangle     created : 10/10/2008   euc
    A parabola associated naturally to an isosceles triangle and a way to generate it through line-intersections.
  52. Generating conics by using the Artzt parabola     created : 10/10/2008   euc
    Use the Artzt parabola of an isosceles triangle to generate various conics through line intersections.
  53. Parabola to circle projectivities     created : 10/10/2008   euc
    Study of a connection of the parabola with the circle-bundles on some chords of it.
  54. Artzt-parabolas connection to outer Steiner ellipse      created : 10/10/2008   euc
    A natural connection of the three Artzt parabolas of a triangle to the outer Steiner ellipse of it.
  55. Three conics naturally associated to a circumconic     created : 10/10/2008   euc
    A natural connection of the three conics and a triangle conic with a given perspector.
  56. Intersection point of Artzt parabola with inner Steiner conic     created : 1/2/2008   euc
    The intersection point of an Artzt parabola of first kind with the inner Steiner conic.
  57. Asymptotic triangle
    Some properties of the asymptotic triangles of a hyperbola.
  58. Asymptotic triangle inverse      created : 22/06/2006, updated : 05/05/2008   euc  
    The inverse construction of a hyperbola from one of its asymptotic triangles.
  59. Autopolar or self-polar triangle
    A triangle possessing a circle with respect to which every side is the polar of the opposite vertex.
  60. Autopolar or self-polar triangle - II      created : 05/05/2008   euc
    A natural way to produce autopolar or self-polar triangles with respect to a conic. Fundamental properties of such systems.
  61. Auxiliary Circle
    The circle with diameter the major axis of an ellipse. Its basic properties.

    B    ^

  62. Barycentric Coordinates      created : 22/06/2007, updated : 10/10/2008   euc
    Definition and first properties of barycentric coordinates with respect to a triangle.
  63. Barycentric Coordinates-II     created : 22/06/2006, updated : 05/05/2008, 10/10/2008, 06/02/2011   euc
    Further properties of barycentric coordinates with respect to a triangle.
  64. Barycentric Coordinates-III     created : 22/06/2006, updated : 05/05/2008, 10/10/2008   euc
    Further properties of barycentric coordinates with respect to a triangle.
  65. Barycentric Coordinates Formulas     22/06/2006, 15/12/09, 5/2/10    euc     
    Some basic formulas involving barycentric coordinates with respect to a triangle.
  66. Barycentrics Projectively
    Barycentric coordinates interpreted through a projectivity between standard models of the projective plane.
  67. Basic Quadrangle Identity
    An identity valid for every quadrangle.
  68. Basic Quadrangle Identity (2)
    A second identity valid for every quadrangle.
  69. Bellows
    Slider-Crank triple bellows mechanism, animated through motor-objects of EucliDraw.
  70. Bellows2
    Slider-Crank quadripartite bellows mechanism , animated through motor-objects of EucliDraw.
  71. Bezier
    Illustration of a user-defined-tool creating a Bezier arc with four control-points.
  72. General Bezier
    Illustration of a user-defined-tool creating a General Bezier arc with N control-points.
  73. Rational Bezier
    Illustration of a user-defined-tool creating a Rational Bezier arc with N control-points and N weights.
  74. Bicentric
    Basic properties of quadrangles that have both inscribed and circumscribed circles.
  75. Bicentric rectangular hyperbolas
    Inverting the circles of a bundle involved with bicentric quadrangles.
  76. Bicentric Loci
    Geometric Loci connected with bicentric quadrilaterals.
  77. Bicorn curve     5/2/2010  euc   
    A curve similar to Napoleon's hat appearing when dynamically modifying the classical Pythagora's figure.
  78. Billiard ball trajectories
    Illustration of a user-defined-tool creating Billiard ball trajectories inside an arbitrary polygon.
  79. Billiard ball closed trajectories in triangles
    Example of construction of a closed billiard ball trajectory inside a triangle.
  80. Billiard ball trajectories in a triangle
    Illustration of a user-defined-tool creating Billiard ball trajectories inside an arbitrary triangle.
  81. Bisecting angle of tangents     created : 22/06/2006, updated : 15/12/2009    euc     
    Bisecting the angle of two tangents to a conic from a point. A theorem of Steiner.
  82. Bisector-I     created : 22/06/2006, updated : 02/02/2011
    The basic figure of bisectors of a triangle and some of its properties.
  83. Bisector-II     created : 22/06/2006, updated : 10/10/2008, 02/02/2011   euc
    The basic figure of bisectors of a triangle and some further properties.
  84. Bisector-III
    Some consequences of the properties examined in the two previous files.
  85. Bisector Cross
    A cyclic quadrangle divided in triangles through its diagonals. A nice cross from the incircles of these triangles.
  86. Bisector Parabola
    A parabola associated naturally to a bisector of a triangle.
  87. Bisector Ratio property
    A property of rectangular hyperbolas characterizing them as locus of vertices of triangles with bisectors of fixed ratio.
  88. Bisector Rectangle
    A rectangle created by the bisectors of two triangles inscribed in the same circle.
  89. Bitangent Conics    created : 31/12/2007, updated : 28/08/2009    euc
    Study of a family of conics and the corresponding homographies preserving its members.
  90. Bitangent rectangular member
    The rectangular hyperbola member of a bitangent family of conics.
  91. Bitangent rectangular member II
    The rectangular hyperbola member of a bitangent family of conics, additional remarks.
  92. Bolzano Weierstrass application
    Applying the theorem of Bolzano and Weierstrass to prove Kroneker's theorem.
  93. Box Fractal
    A fractal resulting by repeatedly dividing a square in 9 equal squares.
  94. Braikenridge Theorem     created : 15/12/2009  euc     
    Braikenridge's theorem: converse of Maclaurin's theorem.
  95. Brianchon's theorem
    Brianchon's theorem on hexagons circumscriptibles in circles.
  96. Brianchon's theorem for pentagons and quadrangles      created : 22/06/2006, updated : 05/05/2008   euc
    Brianchon's theorem specialized for pentagons and quadrangles circumscriptible to circles.
  97. Brianchon's theorem for conics      created : 22/06/2006, updated : 05/05/2008   euc 
    Brianchon's theorem on hexagons circumscriptibles in conic sections.
  98. Bricard exact straight-line mechanism
    Illustration of a linkage-mechanism, transforming circular motion to straight-line motion exactly.
  99. Brocard points, Brocard angle
    Definition and first properties of the two Brocard points and the Brocard angle of a triangle.
  100. Brocard points of pivoting triangles
    Some properties of the triangles created from the basic configuration, defining the Brocard points.
  101. Brocard Ellipse
    A quick introduction to the Brocard ellipse of a triangle.
  102. Brocard point as pivot
    The pedal triangle of a Brocard point and the triangles resulting by pivoting about such a point.
  103. Brocard's second triangle
    Definition and first properties of Brocard's second triangle of a triangle ABC.
  104. Butterfly theorem
    A property of three circle-chords intersecting on the middle of the first one.
  105. Bundle Quadrangle
    Some quadrangles related to circle bundles and their properties.

    C    ^

  106. Carnot's problem
    Finding a polygon by prescribing the middles of its sides.
  107. Carnot's theorem     created : 15/12/2009  euc     
    Carnot's theorem on the intersections of triangle-sides with a conic.
  108. Castillon's problem
    Finding a polygon inscribed in a given conic and having its sides passing through given points.
  109. Castillon's problem in circles
    A special case of the previous. Finding a polygon inscribed in a given circle and having its sides passing through given points.
  110. Catenary
    A remarkable curve created by free hanging catenaries.
  111. Centroid and related remarks
    Some remarks on the centroid of a triangle and questions related to triangle centers.
  112. Centroid (Barycenter or center of gravity)
    A property of centroids using vector operations.
  113. Centroid's basic property
    A property of centroids that relies on vector addition.
  114. Ceva's theorem     created : 22/06/2006, updated : 02/02/2011
    The theorem of Ceva on the coincidence of three lines, definition of Cevians.
  115. Cevian Bisectors
    Definition and first properties of an ellipse defined naturally through the cevians of a point.
  116. Cevians and parallels
    When the parallels to the sides define a triangle perspective to the cevian triangle of a point.
  117. Chasles - Steiner conics     created : 22/06/2006, updated : 05/05/2008, 10/10/2008, 02/02/2011   euc
    The generation of conics through homographic correspondence of two pencils of lines.
  118. Chasles - Steiner conics II     created : 10/10/2008   euc    
    The generation of conics through homographic correspondence of two pencils of lines, additional remarks.
  119. Chasles - Steiner conics III     created : 10/10/2008   euc    
    The inverse procedure to the Chasles-Steiner generation, producing a homographic relation out of a degenerate conic.
  120. Chasles - Steiner line enveloping conics
    The dual to the previous generation of conics as envelopes of lines connecting points on two lines related by a homography.
  121. Chasles - Steiner example     created : 05/05/2008   euc
    Example of generation of a conic by the Chasles-Steiner procedure.
  122. Chebyshev's linkage
    Illustration of a linkage-mechanism, due to Chebyshev, giving four-bar approximate straight-line mechanism.
  123. Circle bundle member contact       created : 12/02/2008   euc
    The circle bundle members which are tangent to a given line.
  124. Circle Bundle Problem      created : 05/05/2008   euc
    A problem related to circle bundles leading to a cubic curve.
  125. Circle Bundles
    Two orthogonal to each other bundles of circles. The intersecting and non-intersecting types of bundles.
  126. Circle Bundle Member
    A distinguished circle bundle member of a distinguished circle bundle.
  127. Circle Bundle of tangent circles
    Two orthogonal to each other bundles of circles. Both containing pairwise tangent circles.
  128. Circle locus related to a circle bundle
    The locus of orthocenters of triangles constructed using a circle bundle.
  129. Circle Bundle Transformation     created : 12/02/2008   euc
    A transformation related to a circle bundle and a given line (direction).
  130. Circle Bundle Transformation to hyperbolas     created : 12/02/2008   euc
    A transformation related to a circle bundle and a given line (direction). The images of non-horizontal lines are hyperbolas.
  131. Circle Bundle Transformation to hyperbolas II     created : 12/02/2008   euc
    A transformation related to a circle bundle and a given line (direction). The images of non-horizontal lines are hyperbolas. Bundles of non-intersecting type.
  132. Circle Bundle Transformation to parabolas     created : 12/02/2008   euc
    A transformation related to a circle bundle and a given line (direction). The images of horizontal lines are parabolas.
  133. Circle Bundle Transformation to parabolas II    created : 12/02/2008   euc
    A transformation related to a circle bundle and a given line (direction). The images of horizontal lines are parabolas. The case of tangential bundles.
  134. Circle Bundle Transformation to parabolas III     created : 12/02/2008   euc
    A transformation related to a circle bundle and a given line (direction). The images of horizontal lines are parabolas. The case of bundles of non-intersecting type.
  135. Circles Similarity Centers       created : 22/06/2006, updated : 27/08/09   euc 
    Location of the similarity centers of two circles and a related problem.
  136. Circular Fractal
    Circular Fractal, driving EucliDraw to its limits.
  137. Circle Tangents Properties       created : 02/02/2011
    Study of certain properties of the tangents to a circle from a point.
  138. Circle Centers       created : 02/02/2011
    Study of the locus of centers of circles simultaneously tangent to two circles.
  139. Circular_Cubics
    Circular cubics, created by the intersection-points of two "lines" of circles (circle-bundles).
  140. Circumcenter
    Circumcenter of a triangle.
  141. Circumcenter Locus
    Circumcenter Locus of a triangle inscribed in an ellipse. Parametric equation of the locus.
  142. Circumcevian triangle
    Triangles defined by the intercepts on the circumcircle of the cevians of a point.
  143. Circumcircle-conjugate (Brocard ellipse)
    The Brocard ellipse of a triangle as the conjugate of the circumcircle.
  144. Circumcircle In Trilinear Coordinates
    The equation of the circumcircle of a triangle in trilinear and barycentric coordinates.
  145. Circumcircle Properties   created : 12/02/2008   euc
    Properties of projections of a circumcircle point on the sides of triangles and quadrangles.
  146. Circumconics Generation     created : 28/08/2009   euc
    Generation of circumconics by secants through the harmonic associates of the pivot.
  147. Circumconics Tangent Generation     created : 12/02/2008   euc
    Generation of circumconics tangent to a given line through a point moving on the given line and associated line intersections. The structure of the set of circumconics which are also tangent to a given line.
  148. Circumconics tangents      created : 22/06/2006, updated : 05/05/2008   euc
    Generation of a circumconic of a triangle as envelope of trilinear polars.
  149. Circumconics Tangent to a Line    created : 12/02/2008   euc
    The structure of the set of circumconics which are also tangent to a given line.
  150. Circumconics of a Trapezium      created : 5/2/2010  euc    
    A picture of the family of conics passing through the vertices of a trapezium.
  151. Circummedial triangle
    The triangle intercepted by the medians on the circumcircle of a triangle.
  152. Circumparabola Generation    created : 12/02/2008   euc
    A simple way to generate a circumparabola through line intersections.
  153. Circumscriptible quadrangle     created : 22/06/2006, updated : 05/05/2008   euc
    Proof of the necessary and suficient condition for a quadrangle to be circumscriptible.
  154. Circumscriptible Construction
    How to construct all circumscriptible quadrangles with given sides.
  155. Circumscriptible Quadrilateral     created : 22/06/2006, updated : 05/05/2008   euc
    Quadrilateral circumsctiptible on circle and a basic property of the intersection of its diagonals.
  156. Circumscriptible Quadrilateral II     created : 22/06/2006, updated : 05/05/2008   euc
    Quadrilateral circumsctiptible on circle. Basic intersection-properties for its diagonals.
  157. Cleaver : bisecting the perimeter of a triangle
    Bisecting the perimeter of a triangle through a line. Variations on a theorem of Archimedes.
  158. Clifford-Cayley Theorem   created : 22/06/2006, updated : 12/12/2009   euc
    Clifford-Cayley Theorem on three linked similar triangles.
  159. Closed polygonal lines
    Closed polygonal lines inscribed in circles and recurring to their start point.
  160. Closed polygonal lines in conics
    Closed polygonal lines inscribed in conics and recurring to their start point.
  161. Complete Quadrangle
    Complete quadrangle, its definition and basic properties.
  162. Common polars
    Points on the major axis of the ellipse have the same polar w.r to it and the auxiliary circle.
  163. Common polars consequences
    The polars of a point w.r. to the ellipse and its auxiliary circle intersect on its major axis.
  164. Complex cross ratio      created : 22/06/2006, updated : 28/08/2009   euc
    Complex cross ratio, its definition and basic properties illustrated through pictures.
  165. Cross ratio of four points on a conic -I
    How the ratio of four points on a conic is defined.
  166. Conchoid of Circle
    Conchoid of Circle, how it is generated mechanically.
  167. Conic characterization     15/12/2009  euc  
    Parameterization of conics through quadratic rational functions. Determination of geometric characteristics.
  168. Conic construction     15/12/2009  euc 
    From two tangents, its contact point with one of the tangents and one of its foci.
  169. Conic equation     5/2/2010  euc     
    A study of the conic equation in the setting of vectors and matrices.
  170. Conic from three points and two tangents
    Construction of a conic passing through three points and having two given tangents.
  171. Conic from three points and two tangents-II
    Construction of a conic passing through three points and having two given tangents, second general case.
  172. Conic Homographies       22/06/2006, 28/08/2009   euc
    On homographies of preserving a conic.
  173. Conic instrument   5/2/2010   
    A file allowing the dynamic change of the six coefficients determining a conic in a cartesian coordinate system.
  174. Conics and similarities
    Conics resulting by a similarity between two circles.
  175. Conics and similarities II
    Conics resulting by a similarity between two circles. A case of non-tangency to the two defining circles.
  176. Conics and similarities III
    Conics resulting by a similarity between two circles. A case of non-tangency to both defining circles.
  177. Conics degenerate     created : 15/12/2009  euc  
    Variations on a set of two lines related to degenerate conics: degenerate or reducible conics.
  178. Families of conics
    A first view on families of conics.
  179. Conics from circles
    Conics as geometric loci of points equidistant from two circles.
  180. Conic intersections with two variable lines     5/2/2010   euc   
    Description of a line defined by intersecting a conic through two other variable lines. Application to the Fregier point.
  181. Conic passing through origin     5/2/2010   euc   
    A picture of the conic passing through the origin. A related family of conics through the vertices of a trapezium.
  182. Conics in homogeneous coordinates     5/2/2010   euc   
    An introduction to the study of projective conics using homogeneous projective coordinates.
  183. Conics as line-envelopes
    Generalization of a well known theorem, representing a conic as line-envelope.
  184. Conics Maclaurin
    A basic method of generation of conics through intersections of varying line-pairs, illustrated in the trivial case of an equilateral triangle.
  185. Conics Maclaurin II      created : 22/06/2006, updated : 28/08/2009   euc
    A basic method of generation of conics through intersections of varying line-pairs, reducing the general case to the trivial one.
  186. Conics Maclaurin III
    The locus of perspectors of Maclaurin conics passing through three points.
  187. Conics Tangent and intersecting      created : 05/05/2008   euc
    Some properties of conics tangent at a point and intersecting at two other points.
  188. Conics defined through rational Bezier curves
    Illustration of a user tool producing arcs of arbitrary conics using rational Bezier curves of degree 2.
  189. Conics through four points geometric      created : 05/05/2008   euc
    Conics passing through four points considered as triangle conics.
  190. Conics and quadratic parameterizations     created : 15/12/2009  euc     
    Generation of conics through quadratic rational functions along two arbitrary lines.
  191. Conjugate diameters homography     5/2/2010   euc   
    A homography defined on every ellipse through the conjugation of its diameters.
  192. Conjugation of inversions
    Conjugation of inversions and its relation to the mid-circle of a circle pair.
  193. Conjugate diameter involution
    Conjugate diameter involution on the points of a conic.
  194. Conjugate triangles
    Conjugate triangles with respect to a conic. Definition and study of their perspectivity.
  195. Conjugacy with respect to conics      created : 28/08/2009   euc
    Conjugacy of points/lines with respect to a conic and some applications.
  196. A problem on inscribed angles
    A problem on inscribed angles on two intersecting circles.
  197. Domain of convexity for quadrangles (entire)
    Solution to the problem: Which points produce with their reflections on the sides of a quadrangle a convex quadrangle.
  198. Coordinates Basics     created : 22/06/2006, updated : 31/01/2008, 28/08/2009   euc
    Basic relations in oblique and normal coordinate systems of the euclidean plane.
  199. Coordinates transform     created : 12/12/2007, updated 05/05/2008, 28/08/2009   euc
    Study of a particular coordinate transformation between two ways to project a point on two lines.
  200. Coordinates transform II     created : 12/12/2007, updated 05/05/2008   euc
    Study of a particular coordinate transformation between two ways to project a point on two lines. A variant on the previous one.
  201. Ellipse of cosine theorem
    An ellipse related to the cosine theorem.
  202. Cosymmedian triangle
    The triangle with vertices the traces of the cosymmedians on the circumcircle of a triangle.
  203. Cross Ratio of four points on a Conic -II     created : 22/06/2006, updated : 31/01/2008   euc
    Definition of the cross ratio of four points on a conic and independence from the good parametrization used. And an application to Haruki's lemma.
  204. Cross Ratio definition
    Definition of the cross ratio of four numbers and cross ratio of four points on a line.
  205. Cross ratio of four concurring lines      created : 22/06/2006, updated 05/05/2008, 02/02/2011   euc
    Definition and properties of cross ratio of four concurring lines.
  206. Four tangents cross ratio
    Cross ratio defined by a intersecting four tangents of a conic by a fifth tangent to the same conic.
  207. Cubic through four points
    Construction of a parametric cubic passing through four given points.
  208. Cubic from two conics     created : 05/05/2008   euc
    Construction of a nodal cubic starting from two tangent and simultaneously intersecting conics.
  209. Cubic, its graphic solution     created : 02/02/2011
    Find the roots of a cubic equation using a parabola and a rectangular hyperbola.
  210. Cubic Reduced     created : 05/02/2008, updated 02/02/2011
    The geometric properties of the cubic in reduced form: y = x3 + px - q and an application to the so-called fundamental invariants of a triangle.
  211. Cubic Symmetry
    The symmetry of the graph of the general cubic function y = ax3+bx2+cx +d about its inflexion.
  212. Cubic With Complex Roots
    Some interesting geometric relations of the cubic y = ax3+bx2+cx +d when it has two complex conjugate roots.
  213. Cut under a given angle
    With a line cut another line or circle under a given angle.
  214. Cyclic quadrilateral
    Some elementary properties of the cyclic quadrilateral.
  215. Maximizing property for cyclic quadrangles
    Cyclic quadrangles have maximum area among all quadrangles with the same sides.
  216. Cyclic quadrilateral from projective viewpoint
    How the general projective properties of a quadrangle specialize in the cyclic case.
  217. Cycloid
    Well known curve, produced by a point attached to a wheel moving on a line.

    D    ^

  218. Delone Linkage
    Delone slider-crank mechanism for tracing ellipses.
  219. De Longchamps ellipse   created : 10/10/2008   euc    
    De Longchamps ellipse of a triangle.
  220. Deltoid   created : 10/10/2008   euc    
    Deltoid naturally associated to a triangle.
  221. Deltoid basic overloaded picture   created : 10/10/2008   euc    
    An overloaded picture of the Deltoid associated to a triangle.
  222. Derivative triangle of another triangle  created : 10/10/2008   euc    
    An equilateral triangle naturally associated to a triangle and its deltoid.
  223. Desargues theorem      created : 12/12/2007, 28/08/09, 10/01/12   euc
    Desargues theorem on perspective triangles.
  224. Desargues theorem II
    Desargues theorem on perspective triangles applied repeatedly to generate a line of perspective triangles.
  225. Desargues involution theorem
    Desargues theorem on involutions defined on lines through bundles of conics.
  226. Desargues involution theorem-II
    Desargues theorem on involutions defined on lines through bundles of conics. Second case.
  227. Desargues involution theorem-III
    Desargues theorem on involutions defined on lines through bundles of conics. Third case.
  228. Desargues involution theorem, Complex version
    Desargues theorem on involutions defined on lines through bundles of conics. Valid in the complex projective plane.
  229. Diameter Property       22/06/2006, 3/01/10   euc    
    Two properties of the diameter of a circle related to angles and products of segments.
  230. Director circle of an ellipse
    An interesting circle from which an ellipse is viewed under a right angle.
  231. Directrix property     created : 15/12/2009  euc   
    A basic directrix property of conics.
  232. Division Problem
    A fairly general problem and its first instances.
  233. Diocles cissoid curve
    Geometric construction of the cissoid of Diocles.
  234. Diocles cissoid curve (2)
    Cissoid of Diocles in terms of implicit functions.
  235. Dividing the sides of a triangle
    The still unsolved problem of counting all nodal points of a triangle's side-division.
  236. Division in ratio     created : 15/12/2009  euc    
    To draw a line through a point intersecting two given lines and forming segments of given ratio.
  237. A distance function
    Graph of the distance function between parallel tangents of a circle and an ellipse.
  238. Domain of convexity
    Solution to the problem: Which points produce with their reflections on the sides of a regular hexagon another convex hexagon.
  239. Domain of convexity for quadrangles
    Solution to the problem: Which points produce with their reflections on the sides of a quadrangle another convex quadrangle.
  240. Droz Farny
    Droz Farny theorem on collinearity of points cut on sides of a triangle by two orthogonal lines.
  241. Duality in projective spaces
    The duality principle in projective geometry, interchanging points with lines and joins with intersections.
  242. Dudeney's dissection
    Dudeney's dissection of an equilateral triangle into a square.

    E   ^

  243. Eight Point Circle
    A circle intimately related to an orthodiagonal quadrangle. Starting point for the study of these quadrilaterals.
  244. Eindvardts Parabolograph
    A link-gear mechanism to draw parabolas and their tangents.
  245. Elations      created : 12/12/2007, 28/08/2009   euc
    A special kind of perspectivities, concentrating all their fixed points on a line.
  246. Elation Decomposition      created : 12/12/2007, 28/08/2009   euc
    Decomposition of an elation in a product of two harmonic perspectivities(homologies) and related questions.
  247. Eleven Points Conic
    A remarkable conic associated to four points and a line in general position.
  248. Eleven Points Conic II     created : 28/08/2009   euc
    The eleven points conic generated by harmonic conjugates with respect to two angles.
  249. Ellipse      created : 22/06/2006, updated 11/08/2009   euc
    The very basic facts about an ellipse.
  250. Ellipse from eccentricity     created : 15/12/2009  euc     
    Properties of ellipse deduced from its definition using a focus and a directrix.
  251. Ellipse as Envelope     created : 28/08/2009   euc
    A very simple generation of the ellipse as envelope of lines.
  252. Elliptical caustic
    Generated instantly as a geometric locus, through the appropriate tool of EucliDraw.
  253. Ellipse characterization     created : 15/12/2009  euc    
    Parameterization of ellipse through quadratic rational functions. Determination of geometric characteristics.
  254. Ellipse Construction
    Ellipse Construction through a fairly simple recipe.
  255. Ellipse Construction (2)
    Another ellipse construction through a more simple recipe.
  256. Ellipse Evolute
    The curve enveloping the normals of an ellipse. Geometric construction of curvature centers.
  257. Ellipse from circles
    Ellipse defined as locus of equidistant points from two circles inside one another.
  258. Ellipse Generation
    Ellipse generation through a simple mechanism using two concentric circles.
  259. Ellipse Generation from circle and point     created : 28/08/2009   euc
    Ellipse generation by parallel projections of circle-points.
  260. Enclosing frame of rotated rectangle
    Studying the enclosing frame of a rectangle rotating about its center.
  261. Roulette 3 to 1
    Tracing a point on a circle rolling on another circle.
  262. Equal bisectors property of rectangular hyperbolas
    Rectangular hyperbolas as loci of vertices of triangles having two equal bisectors (internal and external).
  263. Equal circles at vertices
    Some properties of the figure resulting by constructing equal circles at the vertices of a triangle.
  264. Equal circles conic
    Definition and first properties of conics defined by taking equal circles at the vertices of a triangle.
  265. Equal segments on a triangle
    Solution of a simple construction problem with a triangle.
  266. Equal segments on a triangle-2
    A more structural solution of the previous problem.
  267. Equilateral and Stewart     created : 02/02/2011
    A simple application of Stewart's theorem to the equilateral triangle.
  268. Minimization problem from Euclid
    Minimization problem for the area of a triangle originated from Euclid.
  269. Euler Circle and Line
    Euler Circle and Line of a triangle.
  270. Euler Circle
    Some elementary properties and a connexion to the DeLongshampes point.
  271. Four Euler circles      created : 22/06/2006, updated : 31/01/2008   euc
    The Euler circles of the four "partial" triangles formed from the vertices of a quadrilateral.
  272. Euler Circle Property
    A property used to animate the rectangular hyperbolas circumscribed on a triangle.
  273. Euler Conics
    The two dual triangle conics related to the Euler line.
  274. Euler's relation between in/circum- radius      created : 10/10/2008  euc
    The simple relation between the inradius and circumradius of a triangle.
  275. Eye curve     5/2/2010   euc   
    A bicircular quartic generated through a simple geometric locus which is also the inverse of a related ellipse.

    F   ^

  276. Fagnano's problem
    Solution of Fagnano's problem based on reflexions on the sides of a triangle.
  277. Fagnano's problem II
    Another solution of Fagnano's problem based on reflexions on the sides of a triangle.
  278. Fermat's triangle-point      05/02/2010, 02/02/2011
    Fermat's point of a triangle, minimizing the sum of distances from its vertices. Its trilinear coordinates derivation.
  279. Fermat_Double
    A remarkable phenomenon on repeating the construction of equilaterals on the sides of a triangle.
  280. Feuerbach's theorem
    A proof of Feuerbach's theorem, on the tangency of the Euler-circle with the tritangent circles of a triangle.
  281. Fibonacci
    A visual proof of an area property of the sequence of Fibonacci integers.
  282. A five bar linkage
    A five bar linkage animated through motor-objects of EucliDraw.
  283. Locus of focal points of ellipses       created : 2/02/2011
    A special family of ellipses and the locus of their focal points
  284. Focal points on an ellipse
    An interesting property of the points of an ellipse.
  285. Four equal circles        created : 12/02/2008   euc
    Three equal circles centered on a fourth equal to the three have their second intersection points on an equal fifth circle.
  286. Four-points
    Definition and some elementary projective properties of four-points.
  287. Four-points circumscriptible
    Four-points inscribed in circles. Elementary properties.
  288. Conics through 4 pts and tangent to a line
    Given four pts and a line in general position, there are no or two conics passing/tangent to the line.
  289. Conics through 4 pts and tangent to a line-II
    Given four pts in general position, and a tangent passing through one of them.
  290. Four tangent circles
    An interesting point and four tangent circles naturally constructed in a cyclic quadrilateral.
  291. Fregier      created : 22/06/2006, updated : 30/08/2009   euc
    Fregier's theorem on conic-involutions, defined by pairs of orthogonal lines through a fixed point of the conic.
  292. Fregier Involutive homographies
    The general behaviour of involutive homographies, preserving a conic.
  293. Products of involutive homographies
    How to find geometrically the composition of involutive homographies, preserving a given conic.
  294. Fregier Polar
    The polar line of the Fregier involution by orthogonals and its practical construction.
  295. French trio
    A combination of three famous theorems to explore the structure of symmetric hexagons.
  296. Fundamental invariants of a triangle        created : 02/02/2011
    Definition and introductory discussion of the fundamental invariants of a triangle.

    G   ^

  297. Gauss Bodenmiller theorem
    The circles having diameters the diagonals of a complete quadrilateral are coaxal.
  298. Gear Curve
    Parametric form to generate a gear disc with variable radius, teeth height and number of teeth.
  299. Geometric Series
    Proof of the convergence of a geometric series by inspecting triangle division.
  300. Gergone Universality
    The universal property of Gergonne's point of a triangle.
  301. GIO-Construction of a triangle       created : 02/02/2011
    Around the construction of a triangle from its elments G(barycenter), I(incenter) and O(circumcenter).
  302. Glide-Reflexion        created : 22/06/2006, updated : 31/01/2008   euc
    Definition of glide-reflexion and study of the composition of three reflexions on the respective sides of a triangle.
  303. Good Parametrizations of Conics
    The most "natural" parametrizations of conics, generalizing stereographic projection and establishing homeomomorphism with the projective line.
  304. Inverse of a good parametrization
    The inverse of a good parametrization i.e. rational parametrizations of conics.
  305. Graphical solution     created : 15/12/2009  euc     
    Graphical solution of the quadratic equation and its generalizations.
  306. Graphical solution of the cubic     created : 15/12/2009  euc     
    Graphical solution of the cubic equation.
  307. Graphical solution of the quartic     created : 15/12/2009  euc     
    Graphical solution of the quartic equation.
  308. Tangent at a point of a function-graph
    Tangent at a point of a function-graph and illustration of the dynamic-magnification capability of EucliDraw.
  309. Graph of a function as a geometric locus
    Example of use of the tool [Length-Gauge] of EucliDraw.

    H   ^

  310. Harmonic Conjugate points   05/05/2008, 05/2/2010   euc    
    Definition of this basic relation for a quadruple of points on a line. A basic figure connected with them.
  311. Harmonic Bundle of four lines       22/06/2006, updated : 31/01/2008, 05/05/2008   euc
    Definition, basic properties and some implication of this basic figure of a quadruple of concurring lines.
  312. Harmonic bundle problem      05/05/2008, updated : 14/12/2009   euc 
    A problem easy to solve using properties of harmonic bundles.
  313. Harmonic common conjugates     created : 05/05/2008   euc
    The pair of points simultaneously harmonic conjugate to two other pairs.
  314. Harmonic Perspectivity  created : 05/01/10, updated : 10/01/12  
    Illustration of one of the simplest ways to define a projectivity of the plane.
  315. Harmonic Perspectivity Example       created : 22/06/2006, updated : 30/08/2009   euc
    Construction of a particular harmonic perspectivity associated with four points in general position.
  316. Harmonic Quadrangle     created : 22/06/2006, updated : 30/08/2009   euc
    Cyclic quadrangles such that their vertices, considered as complexes, satisfy (ABCD)=-1.
  317. Hart lemma
    Hart's lemma towards a proof of the famous "Great Poncelet Theorem".
  318. Hart Linkage
    Hart's exact straight-line mechanism, realized through motor-objects of EucliDraw.
  319. Hart Exact straight line mechanism
    Another exact straight-line mechanism, due to Hart. Its geometric background.
  320. Haruki's theorem
    Haruki's theorem on four intersecting circles of two bundles.
  321. Haruki's theorem (2)
    Haruki's theorem on four intersecting circles of two bundles, another instance.
  322. Hexadivision
    Dividing the perimeter of a triangle through equidistant points.
  323. Hexadivision Symmetric
    The equilateral hexagon of minimum area inscribed in a triangle.
  324. Hexagon equilateral inscribed
    The center of the equilateral hexagon of minimum area inscribed in a triangle.
  325. Hexagons symmetric repeated
    A sequence of symmetric hexagons generated by a symmetric one.
  326. Hippopede of Proclus     5/2/2010   euc     
    A bicircular quartic appearing when studying the chords of a circle through a point.
  327. Hippopede of Proclus II     5/2/2010   euc     
    A generation of the previous quartic as envelope of circles passing through a point.
  328. Hippopede Generalization     5/2/2010   euc   
    A picture of a curve generalizing a way of generation of the Hippopede of Proclus.
  329. Homofocal conics
    Orthogonality of an ellipse and a hyperbola with common focal points.
  330. Line Homography       22/06/2006, updated : 31/01/2008, 13/12/2009   euc  
    A basic relation between the points of two lines entering in many problems of projective Geometry.
  331. Line Homography Example
    A basic example of the homographic relation between the points of two lines.
  332. Homography Axis
    Study of the homography axis of a homography preserving a conic. Another viewpoint for Pascal's axis.
  333. Conic-homography 3+3
    Behaviour of a conic-homography defined by three points and their images.
  334. Homothetic Conics
    Some questions related to two homothetic conics.
  335. Homotheties composition     created : 02/02/2011
    Compositions of homotheties. The basic facts and some applications.
  336. Homothety property
    A trivial property for the intersection-points of two families of parallel lines.
  337. Hyperbola     created : 22/06/2006, updated : 30/08/2009   euc
    Basic facts about hyperbolas.
  338. Hyperbola from middles     created : 02/02/2011
    A property of hyperbolas related to lines tangent to their auxiliary circle.
  339. Hyperbola as envelope of lines     created : 28/08/2009   euc
    Hyperbola generation by parallel projections of circle-points.
  340. Hyperbola as envelope of lines II     created : 28/08/2009   euc
    Hyperbola generation by parallel projections of circle-points, generalization by homotheties.
  341. Hyperbola as a graph     created : 05/05/2008   euc
    Representing a hyperbola as the graph of a function.
  342. Hyperbola Asymptotics     created : 22/06/2006, updated : 31/01/2008   euc
    Basic facts about asymptotic lines of hyperbolas.
  343. Hyperbola Asymptotics property
    Some more facts about asymptotic lines of hyperbolas.
  344. Hyperbola through its asymptote     5/2/2010   euc   
    Constructing the hyperbola passing through the vertices of a trapezium and having one given direction for asymptote.
  345. Hyperbola characterization     created : 15/12/2009  euc     
    Parameterization of hyperbola through quadratic rational functions. Determination of geometric characteristics.
  346. Hyperbola Construction
    Using a typical property of hyperbolas to construct them.
  347. Hyperbola through equal segments     created : 05/05/2008   euc
    Generating a hyperbola by tracing equal segments.
  348. Hyperbola From Ellipse
    Creating a hyperbola as image of an ellipse under a projectivity.
  349. Hyperbola From Rectangular
    Derivation of the general hyperbola from the standard one (graph of y=1/x) through an affinity.
  350. Hyperbola Generation
    Creating a hyperbola as image of a circle under a projectivity.
  351. Hyperbola Generation II
    Creating a hyperbola as image of a circle under a projectivity. A second variant.
  352. Hyperbola generated by line intersections     created : 28/08/2009, updated : 12/12/2009   euc    
    Hyperbola generation by intersecting radii of moving points on circles.
  353. Hyperbola with given asymptotics     created : 28/08/2009, updated : 12/12/09   euc    
    Hyperbola with given asymptotics and passing through one given point.
  354. Hyperbola Property
    A property of hyperbolas related to its asymptotic lines.
  355. Hyperbola as locus of points     created : 28/08/2009   euc
    Hyperbola generation as locus of points intersecting segments ending on fixed lines in fixed ratio.
  356. Hyperbola Property with middles
    A property of hyperbolas as locus of middles of segments through fixed point intersepted by two lines.
  357. Hyperbola Property with parallels       created : 22/06/2006, updated : 05/05/2008   euc
    Creating a hyperbola by varying to parallel lines. A subject with interesting applications.
  358. Hyperbola as locus of points       created : 22/06/2006, updated : 05/05/2008   euc
    Hyperbola as locus of points on segments through fixed point A and intersecting these segments in fixed ratio.
  359. Hyperbola Rectangular
    Basic properties of rectangular hyperbolas, having orthogonal asymptotic lines.
  360. Hyperbola related to equilateral       created : 11/08/2009, updated : 12/12/09   euc    
    A special hyperbola generated by secants and tangents at points symmetric with respect to a diameter.
  361. Hyperbola with respect to its asymptotics       created : 22/06/2006, updated : 31/01/2008   euc
    The equation of the hyperbola with respect to its asymptotic lines. The family generated by the hyperbola and its asymptotes.
  362. Hypocycloid
    A curve generated by a fixed point on a circle rolling inside another circle.
  363. Hypotrochoid
    A curve generated by a point rigidly attached to a disc rolling inside circle.

    I   ^

  364. Implicit function
    A curve whose points are determined through an equation.
  365. Incircle of a triangle      02/02/2011
    Basic facts on the incircle and Excircles of a triangle.
  366. Incircle dual
    A remarkable ellipse, representing the dual of the incircle relative to a triangle.
  367. Incircle in trilinears     5/2/2010   euc   
    Calculation of the incircle and the excircles of a triangle in trilinear (and/or barycentric) coordinates.
  368. Incircle tangents       created : 22/06/2006, updated : 05/05/2008   euc
    Generating the tangents of the incircle of the equilateral triangle through trilinear polars.
  369. Incircle tangents II
    A trivial remark on the previous topic, useful in other topics.
  370. Inclined Nodes
    A remarkable curve, generated geometrically and having a simple equation.
  371. Inconics in trilinears     5/2/2010   euc   
    The form of equations in trilinear coordinates of conics tangent to all three sides of a triangle. The role of perspector. The duality to circumconics.
  372. Inconics tangents        created : 22/06/2006, updated : 05/05/2008   euc
    Generation of an inconic of a triangle as envelope of trilinear polars.
  373. The triangle of the radical axes of three circles
    Resembles to the tritangent circles of a triangle ... but not exactly.
  374. Conics inscribed in triangles        created : 22/06/2006, updated : 05/05/2008   euc
    Conics inscribed in triangles generated as envelopes of lines.
  375. Inscribed Squares        22/06/2006, 02/02/2011
    The 6 inscribed squares of a generic quadrangle.
  376. Triangle inscribed in triangle
    How to inscribe a triangle into another triangle.
  377. Triangle inscribed in triangle (2)
    How to inscribe a triangle into another triangle. The phenomenon of the pivot point.
  378. Triangle inscribed in triangle (3)
    How to inscribe a triangle into another triangle. Rotating the inscribed about the pivot point.
  379. Inscriptible Construction
    Construct a quadrangle inscriptible in a circle through its side-lengths.
  380. Intersection of a conic with a line  created : 02/01/12  
    Construct the intersection points of a conic given by five points and a line.
  381. Inverse pedal triangles        created : 22/06/2006, updated : 31/08/2009   euc
    Inverse pedal triangles with respect to an Inversion on a circle.
  382. Lengths and inversion
    Length comparison of segments with inverse end-points.
  383. Inversion
    Introduction to the inversion transformation and its elementary properties.
  384. Inversion generalization     created : 15/12/2009  euc     
    Generalization of the inversion in the projective plane, with respect to some conic.
  385. Inversion Interchanging
    The construction of the inversion-circle, interchanging two given circles.
  386. Product of inversions
    The effect of the product of two inversions on a circle orthogonal to the two inversion-circles. Projective viewpoint.
  387. Inversion Property
    A property of the inversion with respect to a circle used to prove Archimedes' Circles theorem.
  388. Inversion as involution
    The projective aspect of inversion when restricted to an invariant circle.
  389. Inverting on a family of circles
    The envelope of circles created by inverting a circle with respect to members of a family of circles.
  390. Inverting to canonical bundles
    Inverting an hyperbolic bundle and its orthogonal to concentric circles and their radial lines.
  391. Inverting to canonical bundles-2
    Inverting a bundle of pairwise tangent circles to a bundle of parallel lines.
  392. Inverting to equal circles
    A property of the inversion circle of two circles, related to the possibility to invert to two equal circles.
  393. Homographic involutions and circle bundles
    A simple and nice interpretation of a homographic involution through a circle bundle. Connection of bundle-characteristics with fixed points of involution.
  394. Homographic involutions and circle bundles II
    A procedure which, in some sense, is inverse to the one described in the previous reference.
  395. Homographic involutions and circle bundles III
    A slightly more general than the previous precedure of definition of an involutive homography.
  396. Basic example of involutive homography       created : 22/06/2006, updated : 31/01/2008   euc
    The involutive homography defined on a line through the intersections with the sides of a complete quadrilateral.
  397. Basic example of involutive homography-II
    The involutive homography defined on a line through the intersections with the sides of a complete quadrilateral. Determination of fixed points.
  398. Basic example of involutive homography-III
    The involutive homography defined on a line through the intersections with the sides of a complete quadrilateral. Just another figure.
  399. Product of homographic involutions
    The product of two homographic involutions and its characteristics expressed in terms of the data of the factors.
  400. Product of two homographic involutions        created : 22/06/2006, updated : 31/08/2009   euc
    The general case of a product of two harmonic perspectivities of the projective plane.
  401. Involutive homography       created : 22/06/2006, updated : 31/08/2009   euc
    Homography on a conic which is inverse to itself. The basic properties.
  402. Isodynamic points      05/02/2005,02/02/2011
    Isodynamic points, Lemoine and Brocard axes of a triangle.
  403. Isogonal on the bisector     5/2/2010   euc   
    A study of the isogonality with respect to a bisector of a triangle.
  404. Isogonal conjugation
    How to define the transformation of isogonal conjugation with respect to a triangle.
  405. Isogonal conjugation (2)
    Application of the isogonal conjugation on the common tangents of three circles.
  406. Isogonal as quadratic
    The isogonal transformation as a quadratic transformation defined by a family of conics.
  407. Isogonal generalized       created : 22/06/2006, updated : 31/01/2008, 05/05/2008    euc
    The general "isogonal" transformation, associated to a triangle and a point, as a quadratic transformation defined by a family of conics.
  408. Isogonal of circumcircle      created : 05/05/2008   euc
    Application of the isogonal conjugation to points on the circumcircle of the triangle of reference.
  409. Isogonal of parabola      created : 05/05/2008   euc
    Lines whose isogonals with respect to the triangle of reference are circumscribing parabolas.
  410. Isogonal of parabola II      created : 05/05/2008   euc
    Lines whose isogonals with respect to the triangle of reference are circumscribing parabolas. Further properties.
  411. Products of isometries (2)
    Determination of the composition of rotations about the vertices of a quadrangle + a translation.
  412. Isosceles intersection
    A characteristic property of the circumcenter of an isosceles triangle.
  413. Isosceles property
    A property of the isosceles representing a clue on the topic of trilinear polars.
  414. Isotomic as quadratic
    Representation of the isotomic conjugation as a quadratic transformation relative to a family of conics.
  415. Isotomic Chart     created : 05/05/2008   euc
    Diving into the incidence relations of trilinear polars of isotomic points with respect to a triangle and related lines and conics.
  416. Isotomic Conic of a Line      created : 22/06/2006, updated : 05/05/2008, 10/10/2008    euc    
    A recipe to obtain conics from lines using trilinear coordinates and isotomic conjugation.
  417. Isotomic General     created : 05/05/2008   euc
    Some properties of the generalized isotomic transformations with respect to a triangle.
  418. Isotomic of circle     created : 05/05/2008   euc
    The line whose isotomic with respect to a triangle is its circumcircle.
  419. Isotomy on median      created : 22/06/2006, updated : 31/01/2008   euc
    Determination of the isotomic transformation w.r. to a median in trilinears.
  420. Isotomic Chart     created : 05/05/2008   euc
    Relations between the trilinear polars and other remarkable lines and their isotomic conjugates.
  421. Iterative
    What is an iterative procedure for the determination of roots of equations.
  422. Iterative applied to cos(x)
    Explanation of a curious phenomenon when playing with a calculator supporting the cos(x) function.

    J   ^

  423. Jerabek hyperbola
    The Jerabek rectangular hyperbola of a triangle.

    K   ^

  424. Kepler Problem
    A problem of Kepler asking the construction of the maximal barel inscribed in a sphere.
  425. Kiepert hyperbola
    The Kiepert rectangular hyperbola of a triangle.
  426. Koch Snowflake curve
    A fractal curve obtained by a simple construction, starting with an equilateral triangle.
  427. Koch Snowflake generalized curve
    A slightly more general than the previous one fractal curve. Construction based on arbitrary triangle.
  428. Koch Snowflake generalized curve-2
    A slightly more general than the previous one fractal curve. Construction based on arbitrary rectangle.
  429. The Koch Snowflake general curve
    Illustration of a fairly general procedure to construct Koch curves, out of arbitrary polygons and arbitrary broken lines.
  430. Yet another Koch fractal curve
    A more efficient general procedure to construct Koch fractal curves, out of arbitrary polygons and arbitrary broken lines.

    L   ^

  431. Laisant Trisector
    A mechanical device to trisect angles.
  432. Lemniscatoids
    Four-bar crossed-crank linkage for drawing Lemniscatoids.
  433. First Lemoine circle     created : 15/12/2009  euc     
    Circle formed by the intersections with the sides by parallels through the symmedian point.
  434. Second Lemoine circle of a triangle        created : 05/10/2008, updated : 12/12/09   euc    
    A circle formed by intersection points with anti-parallels to the sides of a triangle
  435. Levy Tapestry
    A fractal producing a nice tapestry motif. Using a custom tool of EucliDraw.
  436. Limacon       created : 22/06/2006, updated : 31/01/2008   euc
    First properties of Pascal's limacon.
  437. Limacon II       created : 31/01/2008   euc
    Further properties of Pascal's limacon.
  438. Lines     created : 15/12/2009  euc     
    Basic facts about lines and their representations through coordinates.
  439. Line or tangential coordinates     5/2/2010   euc   
    An introduction to line or tangential coordinates which parameterize lines like usual coordinates parameterize points.
  440. Line pencils     created : 15/12/2009  euc     
    Pencils of lines passing through a point. Basic properties and aspects.
  441. Line Homography
    Homography between the points of two lines.
  442. Linear combination of polygons
    Polygon moving while remaining similar to itself and two vertices glide on two fixed lines.
  443. Line Homography axis
    A fundamental property of homographies between points of two lines.
  444. Line Homography axis II
    The true story about the line homography axis. Connexion with Chasles-Steiner conics.
  445. Line equation in trilinear coordinates
    Line equation in trilinear coordinates. The dependence on coefficients.
  446. Line locus from ratio     created : 05/05/2008   euc
    A simple line locus related to the constancy of a certain ratio.
  447. Line Similarity axis
    Just a special case of the previous subject.
  448. Lines of segments
    A natural variation of a segment having its end points on two lines.
  449. Lines of segments II
    Connexion of the previous topic with a parabola and relations to a generalized Thales theorem.
  450. Link Gear Dwell
    Realization of a link gear mechanism through motor-objects of EucliDraw.
  451. Link Point Locus
    The locus of a point on a link, whose end-points move on two circles.
  452. Lissajous curves
    Mechanical generation of the Lissajous curves through motor-objects of EucliDraw.
  453. Locus Concentric
    A simple geometric locus, related to two intersecting circles.

    M   ^

  454. Maclaurin
    The Maclaurin way of constructing conics through intersections of varying line-pairs.
  455. Maclaurin, Chasles, Steiner     created : 02/02/2011    
    Some further discussion on the generation of conics through the Maclaurin - Chasles - Steiner method.
  456. Maclaurin dual     created : 15/12/2009  euc     
    Dual of the Maclaurin theorem on the generation of conics as envelopes of lines.
  457. Maclaurin like generation of conics     created : 10/10/2008   euc    
    A way to generate conics somewhat similar to Maclaurin's.
  458. Maclaurin like generation of conics II     created : 10/10/2008   euc    
    A way to generate conics somewhat similar to Maclaurin's, necessity of a collinearity.
  459. Malfatti       created : 22/06/2006, updated : 31/08/2009, 11/01/2013   euc
    Hart's completion of Steiner's solution of Malfatti's problem.
  460. Mathot
    The Monge/Mathot point (or anticenter) of a circular quadrangle.
  461. Mathot (2)
    The Mathot point of a circular quadrangle. Application of the theorem (proof).
  462. Mathot (3)
    The Mathot point of a circular quadrangle. The full image of the previous theorem.
  463. Maximal cyclic quadrangle     5/2/2010   euc   
    On the maximal quadrangle inscribed in a circle whose diagonals pass through a fixed point.
  464. Maximal Ellipse
    On the maximal, in area, ellipse inscribed in a square. The obvious solution.
  465. Maximal Rectangle
    On the maximal, in area, rectangle inscribed in a circle.
  466. Maximal Rectangle in Ellipse
    On the maximal, in area, rectangle inscribed in an ellipse. Its relation to the axes.
  467. Maximal Polygons in Ellipse
    On the maximal, in area, N-sided polygons inscribed in an ellipse.
  468. Maximal Segment
    Maximizing a segment joining two points on two circles.
  469. Maximal Triangles in Ellipse
    On the maximal, in area, triangles inscribed in an ellipse. Some elementary properties.
  470. Maximal Triangles Properties
    On the maximal, in area, triangles inscribed in an ellipse. Additional properties.
  471. Maximum area
    Maximum area property of inscribed quadrangles. Using shapes and combining them with graphs of functions.
  472. On the medial line of a side of a triangle
    Study of some properties of the medial line of a side of a triangle.
  473. Medial Parabola
    The parabola, wich is tangent to the three sides and the medial line of a side of a triangle.
  474. Median property
    A property of the medians related to some rectangular hyperbolas associated with a triangle.
  475. Median triangle       created : 11/08/2009   euc
    The median triangle of a triangle ABC and its relation to Vecten configuration and Brocard angle of ABC.
  476. Menelaus theorem       created : 22/06/2006, updated : 02/02/2011  
    The theorem of Menelaus on the collinearity of three points on the sides of a triangle, and its projective version.
  477. Menelaus applied 3 times
    Some properties of intersecting lines based on Menelaus theorem.
  478. Menelaus theorem and parallelograms        created : 22/06/2006, updated : 31/01/2008, 12/12/09, 02/02/2011   euc  
    Some applications of the theorem of Menelaus to parallelogram questions and their projective-geometric consequences.
  479. Menelaus deduced from Ceva's theorem
    The proof by Silvester of the equivalence of Menelaus to Ceva's theorem. The part from Ceva to Menelaus.
  480. Mid Circle
    The circle defining an inversion which interchanges a pair of circles.
  481. Mid Circle-2
    The circle defining an inversion which interchanges a pair of circles. The case of two tangent circles.
  482. Mid Circle-3
    The circle defining an inversion which interchanges a pair of circles. The case of a circle inside another.
  483. Mid Circles
    The circles defining an inversion which interchanges a pair of circles. The case of two intersecting circles.
  484. Mid Circles-2
    The circles defining an inversion which interchanges a pair of circles. The case of a circle and an intersecting line.
  485. Minimal Ellipse
    On the minimal, in area, ellipse enclosing a square. The obvious solution.
  486. Minkowski Sausage
    A fractal resulting by repeatedly replacing the sides of a polygon with a broken line.
  487. Mid-Circle preservetion by inversions
    The preservetion of the mid-circle of two circles by inversions.
  488. Minimal Ellipse of a parallelogram
    On the minimal, in area, ellipse enclosing a parallelogram. Reducing to the above obvious solution.
  489. Miquel
    Miquel's theorem on four intersecting circles.
  490. Miquel Dual      created : 31/01/2008   euc
    A sort of dual of Miquel's theorem leading to triangles pivoting about a point.
  491. Miquel (2)
    Miquel's theorem on four intersecting circles. A second instance.
  492. Miquel Pentagon
    Miquel's theorem on the pentagram created from an arbitrary pentagon.
  493. Miquel Point      created : 22/06/2006, updated : 31/01/2008   euc
    Miquel's pivot point, Miquel's theorem on four intersecting lines and consequences for parabolas.
  494. Miquel Triangle
    Similarity of some triangles in the Miquel configuration.
  495. Mittenpunkt X(9)
    Some properties of the triangle center X(9) and a relation to triangle center X(84).
  496. Mixtilinear circles     created : 31/01/2008   euc
    Definition and basic property of mixtilinear circles of a triangle.
  497. Moebius involution     created : 15/12/2009  euc    
    An example of a Moebius involution related to the ratio of two segments on a line.
  498. Moebius transformations       created : 11/08/2009   euc
    Definition and basic properties of Moebius transformations.
  499. Moebius diagram of a triangle       created : 11/08/2009   euc
    Definition and basic properties of the Moebius diagram of a triangle configuring classes of iso-brocardian triangles.
  500. Moebius transformation preserving a circle
    Properties of such a transformation, related to circle-bundles.
  501. Moon Motion
    Simplified absolute motion of the moon. Considering movement in elliptic trajectories.
  502. Moving similar polygons
    Moving a polygon similar to itself about a fixed vertex, another vertex gliding on a line.

    N   ^

  503. Nagel Point of a triangle     05/02/2010, 02/02/2011
    Definition and basic property of the Nagel point of a triangle.
  504. Napoleon triangle(s)
    Definition of the Napoleon triangle(s) of a triangle.
  505. Nephroid
    A simple geometric construction of the Nephroid.
  506. Nested Polygons
    How to construct self-repeating schemes using EucliDraw's scheme-sockets tools.
  507. Nested Rotated Polygons
    How to construct self-repeating schemes using EucliDraw's scheme-sockets tools. A variation on the previous.
  508. Newton's way to generate conics     5/2/2010   euc   
    Study of the way Newton generated conics by rotating two angles and taking intersections of their legs.
  509. Newton Line of a quadrangle      created : 22/06/2006, updated : 05/05/2008    euc
    A property studied by Newton: The middles of the three diagonals of a complete quadrangle are collinear.
  510. Newton line for circumscriptible
    The Newton line of a circumscriptible quadrilateral passes through the incenter.
  511. Newton Iterative
    The iterative procedure of Newton to determine roots of equations.
  512. Newton Iterative II
    The iterative procedure of Newton to determine roots of equations. A second example.
  513. Newton line property       created : 11/08/2009   euc
    Two points on the Newton line of a quadrilateral related to the three middles of its diagonals.
  514. Conchoid of Nicomedes
    Definition and construction of the Conchoid of Nicomedes.
  515. Conchoid of Nicomedes (2)
    Conchoid of Nicomedes defined by an implicit function.
  516. Nine Points Conic
    The conic passing through the 6 middles of sides of a complete quadrilateral.
  517. Closed even sided polygons with given middles
    Examples of non-symmetric closed polygons with prescribed middles of their sides.

    O   ^

  518. Object Lattice
    Constructing latices of objects through user-tools of EucliDraw.
  519. Olympiad problem
    A problem from the olympiad contests, where the similarity of triangles plays a particular role.
  520. Orthic triangle      created : 22/06/2006, updated : 10/10/2008    euc    
    The triangle of the traces of the altitudes on the sides of a triangle.
  521. Orthic and Pedal
    Reflecting a line on the sides of a triangle. Study of the triangle of the reflected lines.
  522. Orthic and Pedal II
    Reflecting a line on the sides of a triangle. Additional remarks.
  523. Orthic Axis
    The orthic axis of a triangle, related to the orthic triangle and the Euler line.
  524. Orthocenter      created : 22/06/2006, updated : 10/10/2008    euc    
    Vector properties of the orthocenter of a triangle.
  525. Orthocenter2
    Further properties of the orthocenter of a triangle.
  526. Orthocenter Projections
    Projecting the orthocenter on the line joininig a side-middle with the center of Euler's circle.
  527. Orthocentric Anti-Inversion
    An anti-inversion related to the orthocenter of a triangle.
  528. Orthocycle
    A circle intimately related to a cyclic quadrangle.
  529. Orthocycle and bicentrics
    A perspective using the orthocycle to study the bicentric quadrilaterals inscribed in a circle.
  530. Orthocyclic Characterization
    A characterization of the cyclic quadrilaterals using their orthocycle.
  531. Orthodiagonal quadrangle
    Some basic properties of orthodiagonal quadrangles going back to Steiner.
  532. Orthodiagonal quadrangle-2
    Illustrating some points discussed in the previous file.
  533. Orthodiagonal quadrangle-3
    Some further properties of these remarkable quadrilaterals.
  534. Orthodiagonal quadrangle-4
    And here the hints or proofs of their properties.
  535. Orthodiagonal from Cyclic
    A natural map associating to every cyclic quadrilateral a orthodiagonal one.
  536. Minimal orthodiagonal quadrangle
    Given the length of the diagonals, find the minimal in perimeter orthodiagonal quadrangle.
  537. Orthogonal Diagonals
    Various properties of the orthodiagonal quadrangle, resulting by reflecting a point on the sides of rectangle.
  538. Orthogonal Bundles
    A remark concerning the powers with respect to the circles of circle bundle.
  539. Orthopolar triangles     created : 10/10/2008   euc    
    Triangles related through their corresponding deltoids.
  540. Orthopole of a line     created : 10/10/2008   euc    
    The orthopole line of a point with respect to a triangle.
  541. Orthopole of a line II     created : 10/10/2008   euc    
    The orthopole line of a point with respect to a triangle. Additional properties.
  542. Triangle inscribed in a Rectangular Hyperbola       created : 22/06/2006, updated : 31/01/2008   euc
    The very basic properties of a triangle inscribed in a Rectangular Hyperbola.
  543. Oscillating circle of a conic
    A simple recipe to locate the oscillating circle at a point of a conic.

    P   ^

  544. Pappus
    The theorem of Pappus, generalizing that of Pythagoras, and giving the inspiration for the EucliDraw-logo.
  545. Pappus application      created : 22/4/2013   newa
    An application of Pappus theorem leading, after a few steps, to an autopolar triangle w.r. to the Euler circle.
  546. Pappus Lines
    The theorem of Pappus, on the hexagons with vertices on two lines.
  547. Pappus Lines II
    The theorem of Pappus, a special case to which reduces the general one.
  548. Pappus Self Dual
    The theorem of Pappus, and its characteristic to be self-dual.
  549. Pappus Triangle Construction  created : 22/06/2006, updated : 12/12/2009   euc    
    Constructing a triangle from the three elements (A, a, bA) and related problems.
  550. Parabola      created : 22/06/2006, updated : 11/08/2009   euc
    Definition and the very basic properties of the parabola.
  551. Parabola Problems      created : 22/06/2006, updated : 28/02/2013   update newa
    Discussion of some problems related to parabolas, together with a contribution by Mehmet Kilic.
  552. Parabola properties       created : 12/12/2009   euc    
    Further properties of the parabola.
  553. Parabola Area
    The way Archimedes calculated the area of a sector of a parabola.
  554. Parabola by similar triangles
    A parabola connected to an elementary property of circles.
  555. Parabola through centers    created : 31/01/2008   euc
    Parabola as a locus of centers of circles which are simultaneously tangent to a given line and a given circle.
  556. Parabola Chords     created : 22/06/2006, updated : 31/01/2008   euc
    The basic properties of the chords of a parabola.
  557. Parabola circumscribing trapezium       created : 11/08/2009   euc
    The parabola passing through the four vertices of a trapezium. Basic properties.
  558. Parabola from equal segments     created : 05/05/2008   euc
    Parabola generated by certain equal segments.
  559. Parabola from projections
    A parabola generated by diagonals of quadrangles defined through projections.
  560. Parabola Homographies
    Homographies preserving a parabola and fixing a point of it.
  561. Parabola inscribed in a triangle     created : 05/05/2008   euc
    Study of the parabola inscribed in a triangle.
  562. Parabola inscribed in a quadrangle     created : 05/05/2008   euc
    Study of the parabola inscribed in a quadrangle.
  563. Parabola as envelope of angle-sides    created : 31/01/2008   euc
    Parabola as envelope of an angle-side whose other sides passes through a fixed point and the vertex moves on a line.
  564. Parabola Parameter
    Determination of the parameter of a parabola from the coefficient of the standard equation.
  565. Parabola Projected From Circle
    Creation of a parabola as image of a circle under an appropriate projectivity.
  566. Parabola property
    A property of parabolas concerning the cross ratio, defined by five tangents.
  567. Parabola to oblique axes
    Determination of the focus and directrix of a parabola defined by an equation in oblique axes.
  568. Parabola Symmetries
    Some affine symmetries of the parabola and related properties.
  569. Parabola inscribed in equilateral     created : 05/05/2008   euc
    Equilateral triangles tangent to a given parabola.
  570. Parabola trapezium
    Two additional points on a parabola, defined by two points and its tangents there.
  571. Parallelogram division
    How to divide a parallelogram in four equal, in area, parts, through two lines.
  572. Parallelograms inscribed in ellipse
    General properties of parallelograms inscribed in an ellipse. The maximal in area parallelograms.
  573. Parallelograms inscribed in ellipse-2
    Parallelograms inscribed in an ellipse and having constant area E.
  574. Parallelograms inscribed in ellipse-3
    Parallelograms inscribed in an ellipse, having constant area E and maximal/minimal perimeter.
  575. Parallels Medians
    A case of constancy of direction of a variable line.
  576. Parallels to Ellipse
    Parallels to Ellipse produced through the tool of geometric-loci of EucliDraw.
  577. Parametric cubic
    Construction of the parametric cubic, whose coefficients are four given points.
  578. Pascal     created : 22/06/2006, updated : 11/01/2006, 12/12/2009   euc    
    Pascal's theorem on hexagons inscribed in conics.
  579. Pascal (2)      22/06/2006, 31/01/2008, 31/08/2009, 02/02/10   euc    
    Pascal's theorem on hexagons inscribed in conics. A simple nice proof and a special case.
  580. Pascal-Brianchon duality
    Pascal-Brianchon duality based on polar reciprocity.
  581. Pascal and Euler
    A point on the Euler circle of a triangle controlling some circumconics of the triangle.
  582. Pascal and its inverse
    A question on circumconics of a triangle where the theorem of Pascal as well as its inverse apply.
  583. Pascal on quadrangles      created : 22/06/2006, updated : 05/05/2008    euc
    A special case of Pascal's theorem concerning quadrangles.
  584. Pascal on triangles      created : 22/06/2006, updated : 05/05/2008    euc
    A special case of Pascal's theorem concerning triangles.
  585. Pascal Poncelet Brianchon Meeting
    A figure illustrating three major theorems of projective geometry.
  586. Peaucellier's Inversor
    Well known mechanism to transform circular to rectilinear motion.
  587. Pedal of a triangle       22/06/2006, 31/08/2009, 02/02/10   euc   
    Basic properties of the triangle with vertices the projections of a point on the sides of another triangle.
  588. Pedal problem
    A concurrence problem related to the pedal triangle of a point wiht respect to a triangle.
  589. Pedals of ellipses
    Generation of various pedals through the tool of geometric-loci of EucliDraw.
  590. Pedal Polygons
    Polygons resulting by projecting a point on the sides of a polygon. Study of their area.
  591. Pentadivision
    Construct an equilateral pentagon inscribed in a triangle.
  592. Pentadivision Degenerate
    Construct an equilateral degenerated pentagon inscribed in a triangle.
  593. Pentagon Equilateral
    Construct an equilateral pentagon from some minimal data.
  594. Perrolatz Inversor
    A linkage realizing an inversor with respect to a circle.
  595. Perspective triangles       created : 02/02/2011
    A case of perspective triangles studied in projective coordinates.
  596. Harmonic Perspective triangles       created : 02/01/2012
    Perspective triangles which are both inscribed in a conic.
  597. Perspectivity and perspective triangles       created : 02/01/2012
    Perspective triangles related to projective perspectivities.
  598. Perspectivity       created : 22/06/06, updated : 31/08/009,10/01/12    euc
    A special kind of projectivity related to Desargues theorem.
  599. Perspectivity to the Artzt parabola     created : 10/10/2008   euc    
    On the perspective images of the Artzt parabola.
  600. Perspectivity Through Matrix
    A special kind of perspectivity, defined by a matrix.
  601. Pick
    Pick's theorem on measuring the area of a domain through counting of latice-points.
  602. Philon of Byzantium problem
    Philon's problem on the minimal segment through a fixed point, intercepted by two lines.
  603. Philon's cubic
    A cubic curve intimately related to the problem of Philon.
  604. Cubic equation for Philon's problem
    A cubic equation satidied by the tangens of the slope of Philon's line.
  605. Pirate's treasure
    On treasures and pirates that know geometry.
  606. Piriform Curve
    Definition and construction of the Piriform curves.
  607. Piriforms
    Mechanical production of Piriform curves, through linkages.
  608. Pivots of inscribed triangles       created : 22/06/2006, updated : 31/08/2009    euc
    Points defined naturally when inscribing triangles into other triangles.
  609. 12 pivots
    The 12 pivots of rotation of a triangle inside another triangle.
  610. Points dividing in given ratios
    Construct a triangle, whose sides pass through three given point, and satisfies a certain condition.
  611. Polar       created : 22/06/2006, updated : 02/02/2011
    The polar line of a point with respect to a circle. Definition and first properties. Generalization to conics.
  612. Polar construction
    A simple recipe to construct the polar of a point with respect to a circle.
  613. Polarity and cross ratio
    Preservation of the cross ratio by a polarity with respect to a circle.
  614. Polar properties
    Some properties of the polar line of a point with respect to a circle.
  615. Polar properties-2
    Some further properties of the polar line of a point with respect to a circle. An overloaded picture.
  616. Polar on two lines     created : 02/02/2011
    Definition of the polar line of a point with respect to two lines. The concept of the conjugate polygon with respect to a point.
  617. Polar And Inversion
    Some properties deriving from relations of polars to inversions.
  618. Polar locus
    A remarkable locus of the orthocenter of a triangle directly related to the polar of a point w.r. to a circle.
  619. Polar Construction
    Polar Construction for conics.
  620. A property of polar
    A property of the eccentric angles of intersection points of a polar with the respective conic.
  621. Pollock
    Pollock's configuration.
  622. Pollock (2)
    Pollock's configuration. The statement of the theorem and its proof.
  623. Polynomial Property
    A property of curves defined through graphs of polynomials, generalizing the Newton-Vieta relations of roots.
  624. Poncelet porism
    Poncelet's "great" theorem on polygons inscribed and simultaneously circumscribed about conics.
  625. Poncelet's angle relations    created : 31/01/2008   euc
    Poncelet's relations between angles formed by tangents to conics and focal radii.
  626. Poncelet proof
    An outline of an ingenious proof of "Poncelet's great theorem", based on a lemma by Hart.
  627. Poristic triangles     created : 10/10/2008   euc    
    On one-parameter families of triangles. A simple example.
  628. Power genearalization     22/06/2006, 31/01/2008, 12/12/09, 02/02/10   euc  
    Some properties of conics which reduce to properties of the power in the case of circles.
  629. Power along asymptotic directions     5/2/2010   euc   
    Specialization of the previous for the case of secants in the direction of asymptotes of a hyperbola.
  630. Power along axial direction     5/2/2010   euc   
    Specialization of the previous for the case of secants in the direction of the axis of a parabola.
  631. Projection Polygon
    The polygon of feet of perpendiculars from a point on the sides of another polygon.
  632. Projection Triangle
    The triangle of feet of perpendiculars from a point on the sides of another triangle.
  633. Projective Base     created : 08/08/2007, updated : 31/01/2008   euc
    Four points in general position in the projective plane defining a kind of frame of reference.
  634. Projective Coordinates      31/01/2008, 02/02/10   euc  
    Definition and some examples of use of projective coordinates.
  635. Projectification
    Four points in projective plane defining a kind of frame of reference.
  636. Quadrangles projective collinearity
    A projectivity associated naturally with a general quadrangle.
  637. Projective Line     created : 31/01/2008 euc  
    Definition of projective line and its basic properties.
  638. ProjectivePlane
    Definition of the (standard model of) projective plane.
  639. Projectivities fixing vertices
    Study of properties of projectivities fixing the vertices of a triangle.
  640. Projective Rotations
    Projectivities conjugate to usual rotations with some applications to triangle geometry.
  641. Projectivities
    Definition and main properties of the transformations prevailing in projective geometry.
  642. Resolution to perspectivities
    Resolution of a general projectivity to perspectivities.
  643. Ptolemy
    The proof-idea of Ptolemy's theorem, using inversions.
  644. Pythagora's theorem       11/08/2009, 02/02/10   euc  
    Pythagora's theorem and some simple consequences.

    Q   ^

  645. Quadrangle construction
    Construction of a quadrangle with three equal sides by giving the middles of these sides.
  646. Quadrangle division in four
    How to divide a quadrangle in four equal, in area, pieces, through two lines.
  647. Quadrangles classified modulo affinities
    The moduli space of quadrangles and a representative for each class, modulo affinities.
  648. Quadrangle      11/08/2009, 02/02/10   euc  
    Basic intersection properties on sides and diagonals of a generic quadrangle.
  649. Quadrangle's harmony     5/2/2010   euc   
    A picture illustrating the properties of a general quadrangle related to harmonicity.
  650. Quadratic solving     5/2/2010   euc   
    What are we doing when we solve a quadratic equation?
  651. The quadratic equation in the plane     10/1/2012     
    The quadratic equation in two variables and its reduction to normal form.
  652. Quadratic Transformation
    Illustration of the quadratic transformation, defined through a family of conics.
  653. Quadratic Transformation (2)
    Applying quadratic transformations to lines, to produce conics.
  654. Quadratrix
    The curve used to devide angles in a given ratio.
  655. Quartic
    The (affine) symmetries of a curve created by a polynomial of fourth degree.

    R   ^

  656. Radical Axis of two circles      created : 08/08/2007, updated : 31/01/2008   euc
    Radical Axis of two circles and a plausible property.
  657. Radical Axis of two circles II
    Radical Axis of two circles. A simple property.
  658. Ratio of distances from middle
    Calculation aid for ratios defined by points on a segment.
  659. Reciprocal polar conics
    A duality based on the pole-polar reciprocity with respect to fixed conic.
  660. A property of conjugate lines of a rectangular hyperbola
    A very simple property of conjugate lines of a rectangular hyperbola.
  661. Rectangular Hyperbola
    A rectangular hyperbola defined by a simple relation.
  662. Rectangular Hyperbola from a line     created : 02/02/2011
    A family of rectangular hyperbolas defined by a line in a Cartesian coordinate system.
  663. Rectangular Hyperbola Circumscribed     created : 08/08/2007, updated : 31/01/2008   euc
    Construction-Animation of all rectangular hyperbolas circumscribed on a triangle.
  664. Impossibility to inscribe a rectangular Hyperbola
    Illustration of the typical case where it is impossible to inscribe in a quadrangle a rectangular hyperbola.
  665. Creating parallel chords in rectangular hyperbolas     created : 08/08/2007, updated : 31/01/2008   euc
    Intersecting a rectangular hyperbola with a circle bundle and creating parallel chords.
  666. Rectangular Hyperbola Relation    created : 08/08/2007, updated : 31/01/2008, 02/02/2011   euc
    A relation of the parameters of two lines intersected by a third, expressed through a rectangular hyperbola.
  667. Rectangular hyperbola generation
    A very easy way to construct a rectangular hyperbola.
  668. Rectangular Hyperbolas tangent to four lines
    Construction of the (two) hyperbolas tangent to four given lines (when this is possible).
  669. Rectangular Hyperbola through four points    created : 08/08/2007, updated : 31/01/2008   euc
    Construction of the rectangular hyperbola passing through four non orthocentric points.
  670. Rectangular Hyperbola inscribed in a triangle
    Construction of the rectangular hyperbolas (when existing) tangent to the three sides of a triangle.
  671. Recycler
    A Moebius transformation recycling the vertices of a triangle.
  672. Recycler Homographic
    A Homographic transformation recycling the vertices of a triangle.
  673. Reflexion on conics
    A conic involution resembling the usual reflexion, but defined on a conic.
  674. Reflecting On Polygon Sides
    The composition of reflection on the sides of a quadrangle.
  675. Reflecting On Polygon Sides (2)
    The composition of reflection on the sides of a quadrangle. The final picture.
  676. Reflecting segment ends
    A segment of fixed length gliding on a side of a triangle and its reflexions on the other sides.
  677. Reflexions of line
    A parabola connected with closed billiard trajectories inside a triangle.
  678. Regula Falsi
    The method of "false position" for the determination of the roots of an equation.
  679. Rhodonea
    Curves similar to flowers with any number of petals.
  680. Rolling On Parabola
    Can a wheel roll on a parabola?
  681. Roots
    Construction of qudratic roots of integers. Using scheme-sockets.
  682. Roots (2)
    Construction of qudratic roots of integers. Using scheme-sockets. A second method.
  683. Rotating a right angle into a circle     5/2/2010   euc   
    A study of the rotation of a right angle inside a circle. A related generation of an ellipse.
  684. Rotating the sides of a triangle
    Some figures resulting by rotating the sides of a triangle about their middles.
  685. Rotating the sides of a triangle (2)
    Further study of figures resulting by rotating the sides of a triangle about their middles.
  686. Rotating the sides of a triangle (3)
    Relating Brocard Geometry with figures resulting by rotating the sides of a triangle about their middles.
  687. Rotating Triangle
    How to construct a triangle rotating in its circumcircle.
  688. Rotating Triangle similar
    Some properties related to a triangle, rotating in its circumcircle.
  689. Rotation at point A by an angle w    11/08/2009, 02/02/10   euc  
    Analytic description of the transformation and a simple application.
  690. Rotation product odd
    Calculating the composition of rotations on the vertices of an odd-sided polygon.
  691. Rotation on conics
    A conic involution resembling the usual rotation, but defined on a conic. And a recipe to create many "Poncelet" polygons tangent to two conics.
  692. Rotations On Quadrangle Vertices
    Calculating the composition of rotations on the vertices of a quadrangle.
  693. Rotations On Quadrangle Vertices circum
    Calculating the composition of rotations on the vertices of a circumscribable, on a circle, quadrangle.
  694. Roulette (inner)
    A special case of a roulette resulting by a point rigidly attached to a disc which rolls inside a circle.
  695. Roulette (outer)
    A special case of a roulette resulting by a point rigidly attached to a disc which rolls outside a circle.

    S   ^

  696. S-Triangles      created : 05/05/2008   euc    
    Triangles related by their deltoids.
  697. Salinon
    The geometry of a shield of ancient Greeks.
  698. Seesaw
    The construction of a seesaw on a circular arc.
  699. Geometric Series
    The convergence of the geometric series for q = 1/3, illustrated geometrically.
  700. Sequence of triangles
    A remarkable sequence of triangles created for each triangle and each triangle center of it.
  701. Sewing
    The feed mechanism of a sewing machine, realized through motor-objects of EucliDraw.
  702. Sewing (2)
    Slider-Crank thread and needle guiding mechanism of a sewing machine.
  703. Sierpinsky Fractal
    Application of the tool [Fractal-Socket] of EucliDraw to produce a well known fractal.
  704. Similarity       created : 22/06/2006, updated : 31/08/2009, 02/02/2011    euc
    A first discussion of the similarity transformation together with an interesting application.
  705. Similarly gliding       created : 22/06/2006, updated : 02/02/2011
    Triangles varying with all their vertices on fixed lines while remaining similar all the time.
  706. Similarly rotating       created : 22/06/2006, updated : 02/02/2011
    Triangles with a fixed vertex and varying the other vertices on lines while remaining similar all the time.
  707. Similar Inscribed triangles
    Introduction to similarities through two similar triangles with a common vertex.
  708. Similar Inscribed      created : 22/06/2006, updated : 02/02/2011     
    Introduction to similarities through two similar quadrangles with a common vertex.
  709. Similarity Centers
    Properties of the two similarity centers of two circles exterior to each other.
  710. Simple Geometric Locus
    Simple Geometric Locus, generated by the appropriate tool of EucliDraw.
  711. Simple Geometric Locus (2)
    Simple Geometric Locus, generalizing the previous one.
  712. Simson line of a triangle      created : 22/06/2006, updated : 05/05/2008, 10/10/2008    euc    
    Projecting a point on the circumcircle onto the sides of a triangle.
  713. Simson 3 Lines
    A configuration involving the Simson lines and the Morley triangle.
  714. Simson lines at right angles    created : 27/10/2007, updated : 31/01/2008   euc
    Diametral points on the circumcircle have corresponding Simson lines orthogonal, intersecting on the Euler circle.
  715. Generalized Simson Lines      created : 22/06/2006, updated : 05/05/2008, 10/10/2008    euc    
    Instead of orthogonal projections on triangle sides producing the well known Simson lines, here another recipe.
  716. A parallel to the Simson line      created : 22/06/2006, updated : 05/05/2008, 10/10/2008    euc    
    A line parallel to the Simson line.
  717. The angle between two Simson lines
    A property of the angle of two Simson Lines relative to the same triangle.
  718. Simson Related
    An exercise on Simson lines involving the antiparallel triangle.
  719. Three Simson lines of a triangle
    The triangle formed by the Simson lines of three points on the circumcircle of a triangle.
  720. Generalized Simson Lines Envelope
    The envelope curve of all generalized Simson lines for a fixed angle.
  721. Simson Variant Locus
    Rotate a triangle in its circumcircle and produce a locus with kinetic conditions.
  722. Soliton
    An example of a traveling soliton.
  723. Solitons interfering
    The phenomenon of interference of two traveling solitons.
  724. Somov
    Somov's slider-crank mechanism for tracing Cassinian ovals.
  725. Spiral Tiling
    Algorithm to place M parallelograms in a spiral-like way around the first which serves as prototype for the tiling.
  726. Spiral Tiling Hexagon
    Algorithm to place M regular-hexagonal tiles in a spiral-like way around the first tile.
  727. A property of the square
    A simple property of secants of a square.
  728. Extract the square root
    The classic easy way to extrac geometrically the square root of a number.
  729. Square root mechanism
    A linkage used to extract mechanically the square root of a number.
  730. Squares through
    Given four points A,B,C,D, construct a square, whose sides pass through these points.
  731. Squares through 4_points
    The 6 squares with sides (or their prolongations) passing through four given poins A, B, C, D. The construction is done using the tool of scheme-sockets.
  732. Square construction
    Find the square having two points on opposite sides and two points on the diagonals.
  733. Squares-Dissection   11/08/2009, 02/02/10   euc
    Dissect two squares in pieces and rearrange them into one square.
  734. Square in Square
    Square into another square defines four areas, the two opposite having area sum equal to the other two.
  735. Squares Combination
    The locus of points whose sum of weighted distances from some points remains constant.
  736. Staircase automatic
    Automatic Staircase. Shown the principle which uses the most elementary curve of constant width.
  737. Steiner Chain
    Steiner's theorem on properties of chains of circles, based on inversions.
  738. Steiner Chain
    As before, just another viewpoint.
  739. Steiner Chain
    As before, just a third viewpoint.
  740. Steiner Ellipses       created : 22/06/2006, updated : 31/08/2009    euc     
    Steiner Ellipses of a triangle and how to construct them quickly with EucliDraw.
  741. A theorem of Steiner
    A theorem of Steiner on the generation of ellipses.
  742. A theorem of Steiner (2)
    A theorem of Steiner relating to Pascal's theorem on inscribed hexagons.
  743. Steiner lines
    Lines parallel to Wallace-Simson lines, passing through the orthocenter of a triangle.
  744. Steiner point      created : 08/08/2007, updated : 31/01/2008   euc
    The Steiner point, fourth intersection point of the circumcircle with the outer Steiner ellipse. An easy construction.
  745. Steiner reflected
    Creating the point corresponding to a given Steiner line by reflecting this line on the sides of the triangle.
  746. Stereographic projection
    The classical map serving as prototype for "good parametrizations" of conics.
  747. Stewart's theorem
    Stewart's theorem on the length of Cevians of a triangle.
  748. Strophoid
    A theorem on strophoids taken from an old french book (Aubert-Papelier).
  749. Subnormal of a conic      created : 02/01/12   
    The length of the projection of the normal of a conic on the axis carrying the focal points.
  750. Symmedian line      created : 22/06/2006, updated : 01/05/07, 10/10/08, 12/12/09, 29/03/13    euc    update
    Symmedian line of a triangle through a vertex, Symmedian-point of a triangle, elementary facts.
  751. Symmedian line II
    A bunch of similar triangles defined through a symmedian line.
  752. Symmedian point
    Symmedians and the Symmedian-point of a triangle, as before but other viewpoint.
  753. Symmedian property
    A property characterizing the symmedian lines of a triangle.
  754. Symmedian - Vecten
    Symmedians and their relation to the Vecten configuration of the triangle.
  755. Symmetric hexagons
    Circum/Inscribed conics of symmetric hexagons.
  756. Symmetric hexagons All
    How to classify all symmetric hexagons.
  757. Symmetric Hexagons Special
    Study of a special class of symmetric hexagons.
  758. Symmetric Hexagons Nested
    Study of the special case of symmetric nested (or derived) hexagons through their side-middles.
  759. Symmetric Octagons Nested
    Picture of the special case of symmetric nested (or derived) octagons through their side-middles.
  760. Symmetric triangles with respect to a point
    Pairs of triangles, symmetric with respect to a point and a related conic.
  761. Perspective in a conic inscribable triangles
    Generalization of the previous subject on perspective triangles.
  762. Symmetries On Vertices
    Study of the composition of symmetries on the vertices of a rectangle.
  763. Symmetries On Vertices Even
    Study of the composition of symmetries on the vertices of an even sided polygon.
  764. Symmetries On Vertices Even (2)
    Study of the composition of symmetries on the vertices of an even sided polygon. Symmetric polygons.
  765. Symmetries On Vertices Odd
    Study of the composition of symmetries on the vertices of an odd sided polygon.

    T   ^

  766. Tangent Cuts Circle
    Envelope of tangent-cuts from a circle to two other circles, the three circles building a circle-bundle.
  767. Tangent Cuts Envelope
    Envelope of tangent-cuts from a conic to another conic.
  768. Tangents envelope
    Envelope of contact-joins of tangents from a circle to two other circles, the three circles building a circle-bundle.
  769. Tangent member
    Find the members of a circle bundle that are tangent to a given circle.
  770. All conics tangent to four given lines
    Construction of all conics tangent to four given lines.
  771. Circumcircle of the tangential triangle
    The circumcircle of the tangential triangle, its location on the Euler line and location of X(25).
  772. Tangram
    Tangram realization through EucliDraw.
  773. Terquem General
    Generalization of the Terquem transformation using two special points and a conic.
  774. Tetradivision
    Construct rhombi inside triangles.
  775. Thales theorem
    A first discussion of the classical theorem and its applications.
  776. Thales theorem II      created : 22/06/2006, updated : 01/05/2007, 10/10/2008, 31/08/2009    euc    
    A further discussion of the classical theorem and its applications.
  777. Thales application
    An application of Thales theorem.
  778. A generalization of Thales Theorem      created : 22/06/2006, updated : 31/08/2009    euc    
    Generalization of Thales theorem on the segments cut of on a bundle of lines by two or more parallels.
  779. Thales Parabola      created : 22/06/2006, updated : 31/08/2009    euc    
    A parabola related to two lines and a point not belonging to these lines.
  780. Thales remarks      created : 22/06/2006, updated : 31/08/2009, 02/02/2011    euc    
    Remarks on Thales theorem escalating from the very simple to the more advanced.
  781. Thebault
    Constructing squares on the sides of a parallelogram. Thebault's theorem.
  782. Three angles        created : 11/08/2009   euc
    A property of three angles on a common chord.
  783. Three bar dwell
    Construction of a dwell mechanism.
  784. Three Collinear Points
    An interesting exercise related to a circle.
  785. Three Collinear Points (2)
    As before but more general, involving conics.
  786. Three circles problem
    A problem involving three circles and leading to conic-construction.
  787. Three circles problem (2)
    A conic construction involving three circles and a radical axis.
  788. Three circles problem (3)
    Various curves related to the previous construction, involving three circles.
  789. Three diameters       created : 22/06/2006, updated : 01/05/2007, 10/10/2008    euc    
    Application of the property of Wallace-Steiner lines.
  790. Three lines       12/12/2009, 02/02/10   euc    
    Basic aspects of the configuration consisting of three lines passing through the same point.
  791. Tractrix
    A famous curve, its equation and geometric generation.
  792. Tractrix Mamikon
    Application of the Mamikon device to calculate the area under a tractrix.
  793. Trapezium     12/12/2009, 02/02/10  euc    
    Trapezium's basic properties .
  794. Trapezium II     11/08/2009, 02/02/10   euc      
    Trapezium's some more properties.
  795. Trapezium (2)   11/08/2009, 02/02/10   euc     
    A property of trapezia.
  796. Trapezium Division
    Trapezium's division in two equal (in area) parts, by a line passing through the middle of a non-parallel side.
  797. Producing triangles with the same centroid
    Extending the sides of a triangle proportionally and building triangles with the same centroid.
  798. Triangle Bisectors
    A study of some lines associated with the bisectors of the angles of a triangle.
  799. Two triangle constructions involving bisectors      created : 22/06/2006, updated : 12/12/09    euc    
    Two triangle constructions involving bisectors and a reference to an interesting account of triangle constructions.
  800. Conics circumscribing a triangle-I
    A study of the conics circumscribing a triangle.
  801. Conics circumscribing a triangle-II
    A study of the conics circumscribing a triangle. Just another aspect.
  802. Triangles Circumscribing Parabolas
    The basic properties of triangles having their sides tangent to a parabola.
  803. Triangle Conics       created : 08/08/2007, updated : 31/01/2008, 05/05/2008   euc  
    A discussion of the pairing of conics inscribed/circumscribed on a triangle.
  804. Triangle conics intersections
    The fourth intersection point of two triangle conics. How to find it easily.
  805. Triangle Construction from A, a, rb
    To construct a triangle from A, a and rb.
  806. Triangles with given pivot       created : 12/12/2009   euc    
    Constructing a triangle inscribed in a conic and having a given point as pivot for this conic.
  807. Triangles with given pivot II       created : 12/12/2009   euc    
    Constructing a special triangle inscribed in a conic and having a given point as pivot for this conic.
  808. Triangle Glide Reflexions
    Composition of three reflexions in general position.
  809. Projectivities related to a triangle
    Some basic relations between in/circum-conics of a triangle, cevians and trilinear polars expressed through projectivities.
  810. Triangle Rotations Product
    Composition of three rotations about the vertices of a triangle.
  811. Line of triangles
    Varying a triangle so that its vertices glide on three lines.
  812. Triangle Similarities Product
    Composition of the three naturally defined similarities about the vertices of a triangle.
  813. Triangle Symmetrization
    Transforming an arbitrary triangle to an equilateral in its circumcircle.
  814. Triangles Line
    A kind of line whose points are triangles.
  815. Trilinear Polar       created : 22/06/2006, updated : 05/05/2008, 01/09/2009   euc      
    The line of perspectivity of the Cevian triangle of a point with respect to a triangle.
  816. Find triangle with given trilinear polar       created : 31/01/2008, updated : 01/09/2009   euc      
    Given a conic and a line to inscribe a triangle in the conic having the given line as trilinear polar.
  817. Conics generating parallel trilinear polars       created : 22/06/2006, updated : 05/05/2008   euc  
    Triangle conics whose points have trilinear polars parallel to a fixed direction.
  818. Trilinear Polar Envelope
    A problem of finding the envelope of trilinear polars.
  819. Trilinear projectivity      created : 05/05/2008   euc
    Projectivity generated by a point and a triangle and related to the trilinear polar of the point.
  820. Trilinear coordinates       created : 22/06/2006, updated : 05/05/2008   euc  
    Definition and first properties of the standard trilinear coordinates relative to a triangle.
  821. Trilinears and others      created : 22/06/2006, updated : 12/12/09    euc    
    Relations of trilinear coordinates to some other systems of coordinates associated to a fixed triangle.
  822. Tripole
    The point M with respect to which a line is the trilinear polar for a given triangle.
  823. Trireme
    Picture of a greek trireme, animated through a motor-object of EucliDraw.
  824. Twelve Pivots       created : 11/08/2009   euc    
    The twelve pivots of rotation of a triangle into another.

    U   ^

    V   ^

  825. Van Aubel
    Van Aubel' s theorem on squares on the sides of a quadrangle.
  826. Van Aubel (2)
    Van Aubel' s theorem on squares additional properties and proofs.
  827. Vecten
    The Vecten configuration of a triangle.
  828. Vecten (2)
    More on the Vecten configuration of a triangle.
  829. Vecten (3)
    More on the Vecten configuration of a triangle.
  830. Vecten (4)
    More on the Vecten configuration of a triangle.
  831. Vecten (5)
    More on the Vecten configuration of a triangle.
  832. Vecten Inequality
    A natural inequality connected with the Vecten diagram of a triangle.
  833. Vecten diagram of an equilateral triangle
    A simple case of measurement of the perimeter-ratio of two equilateral triangles.
  834. Vecten tiles
    Glueing Vecten tiles together makes a grid.

    W   ^

  835. Wallace-Simson line
    Some remarks concerning the Wallace-Simson lines of a triangle.
  836. Weighted Sums of Distances
    On the locus of points whose sums of weighted distances from some fixed points remains constant.
  837. Weird Sinus Zoom
    A picture on the use of the Zoom-inside-a-rectangle-tool of EucliDraw.
  838. Woo's Construction
    A simply construction of the incircle of an arbelos.

    X   ^

    Y   ^

  839. Yagci's problem
    Solution of a problem related to the Brocard points of a triangle.
  840. Yates Trisector
    A linkage to trisect angles, originated by Laisant and improved by Yates.

    Z    ^


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