Symmedian

Symmedian (or Lemoine) point of a triangle

[0] The symmedian at C, of a triangle ABC, is defined as the symmetric line of the median from C, with respect to the bisector of C (see Symmedian_0.html for its properties).
[1] The three symmedians of ABC pass through the Symmedian point K of the triangle.
[2] K coincides with the Gergonne point of the tangential triangle A'B'C' of ABC.
[3] Thus, the two triangles ABC and A'B'C' are point-perspective and by Desargues (see Desargues.html ), they are also line-perspective. The corresponding line A*B* is the Trilinear polar of the triangle ABC with respect ot K and is called the Lemoine-Axis of the triangle.
[4] A* is the intersection point of BC and B'C', B* is the intersection point of CA and C'A' etc.
[5] A* is the pole of AA' with respect to the circumcircle of ABC, B* is the pole of BB' etc.
[6] By the duality of the polarity, this implies that K is the pole of the trilinear polar. Hence OK is orthogonal to the trilinear polar.
[7] Line (OK) is the Brocard-Axis of the triangle.
[8] C* is the center of the Apollonian circle of ABC passing through C. Analogous properties hold for B* and A*. Thus the three Apollonian circles have their centers on the Lemoine axis.

The ideas of harmonic bundles of lines, of trilinear polar, of perspective triangles, of symmedian lines, of antiparallels, of Apollonian circles, of tangential triangle, of isodynamic points and of Brocard points are relevant for a full understanding of this subject. Most of them are discussed in the references given below.

Symmedian (or Lemoine) point of a triangle

See Also