[alogo] 1. Isogonal conjugation on a bisector

Consider a triangle ABC and the reflexion of points on a bisector AD of angle A say. This transformation is called isogonal with respect to the bisector AD. The transformation, being a particular reflexion preserves distances and maps lines to lines. In particular a line AX through A is mapped to a line AX' through A, which is fixed by the transformation. Also sides AB and AC are interchanged by the transformation.
Some further properties of this transformation are:
(1) The projections {Y,Z} of X and {Y',Z'} of its transform X' on the sides are on a circle c.
(2) The center of this circle c is on the bisector AD.
(3) The triangles XYZ and X'Y'Z' are equal.
(4) If a line AX is the locus of points such that XY/XZ = k, then its transform AX' is the locus of points X'      such that X'Y'/X'Z' = 1/k.

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Start from (3) showing the equality of triangles XYZ and X'Y'Z'. This is immediate, since by definition the angles at X, X' are equal and complementary to A and the sides XY=X'Z', XZ=X'Y'.
(4) is a consequence of (3). Properties (1) and (2) follow from the fact that YY'ZZ' is an equilateral trapezium, since AY'=AZ etc..

[alogo] 2. Further properties

Consider two isogonal lines AE, AE' of the triangle ABC and take an arbitrary point X on AE and an arbitrary point X' on AE'. Then project X, X' on the sides of the triangle to points {Y,Y'} on AB and {Z,Z'} on AC [Lalesco, p.40].
(1) Triangles XYZ and X'Y'Z' are similar.
(2) The quadrilateral YY'Z'Z is cyclic on a circle c, and {Y'Z, YZ'} are antiparallels.
(3) AE is orthogonal to Y'Z' and AE' is orthogonal to YZ.
(4) The center of the circle c is the middle O of XX'.
(5) The three lines {Y'Z, YZ', XX'} pass through the same point G, whose polar with respect to       the sides {AB, AC} is the orthogonal from A to XX'.

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(1) is valid because quadrangles AYXZ and AY'X'Z' are cyclic, thus angle YAX = YZX, X'Y'Z' = X'AZ' etc..(3) follows also from these angle equalities e.g. X'Y'Z' = YAX and since the sides {AY, X'Y'} are orthogonal the other two sides {XA,Y'Z'} are also orthogonal. (2) follows also from the equality of the angles YY'Z' = YY'X'+X'Y'Z' and YZZ'=YZX+XZZ' but X'Y'Z' = YXZ etc.. (4) follows from the fact that O projects on the middle of YY' on AB and the middle of ZZ' on AC, hence it is the intersection of the two medial lines of these segments of the same circle c.
(5) is proved in section-2 of Pedal.html .
Remark In the aforementioned reference it is also proved that the polar of G with respect to the pair of lines AB, AC passes through the intersection point of lines {YZ, Y'Z'}.

[alogo] 3. Composition of conjugations on bisectors

Consider a triangle ABC and the three reflexions FA, FB, FC with respect to its bisectors at the corresponding angles.
(1) The result of applying the transformations successively to a point X, creating X'=FA(X),
      X''=FB(X'), X'''=FC(X''), defines a cyclic quadrangle XX'X''X''' centered at I, the incenter of       the triangle.
(2) The composition F=FC*FB*FA is the reflexion on the line through I which is orthogonal to AC.
   

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(1) is obvious since reflexions preserve distances, hence all IX, IX', IX'', etc. are equal.
(2) follows from (1). In fact, angle X'XX''' is then complementary to X'X''X''', which in turn has its legs orthogonal to the bisectors IB, IC. By measuring angle it follows that XX''' is parallel to AC. Since then XIX''' is isosceles X and X''' are symmetric with respect to the line from I which is orthogonal to AC.
Remark-1 This property of the composition of reflexions is a special case of composition of several reflexions on lines passing through a point. If the number of different lines is even then the result-composition is a rotation if the number of lines is odd (as is here the case), then the result is a reflexion.
Remark-2 This composition of isogonal conjugation must be distinguished from the isogonal conjugation with respect to a triangle which is another kind of important transformation,
playing an essential role in the geometry of the triangle and being quadratic in nature and transforming lines to conics passing through the vertices of the triangle  (see Isogonal_Conjugation.html ).

See Also

Pedal.html
Isogonal_Conjugation.html

Bibliography

[Lalesco] Lalesco, Trajan. La Geometrie du Triangle. Paris, Jacques Gabay, 1987

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