What is the isotomic conjugate version of the six point circle of isogonal conjugates?

Question

As it is well known, the pedal triangles of a pair of isogonal conjugates in a triangle share a circumcircle.

This is a nice theorem, but is there an analogous version of it for a pair of isotomic conjugates?

Thanks.

Answer 1

The most we can say is that the six points resulting from isotomic conjugation will lie on one conic.

To see this, let $I,I_AI_BI_C$ be the incenter and excentral triangle of $ABC$. Then let $G, G_AG_BG_C$ be the centroid and anticomplementary triangle of $ABC$.

I will refrain from copying the proof of another, which you can view here in a post/article: http://artofproblemsolving.com/community/q4h621725p3716734

Anyway, the article linked to discusses how isogonal conjugation is a reciprocation in a projective coordinate system where $I,I_A,I_B,I_C$ form the basis of the system, and similarly for isotomic conjugation. Thus we can determine a projective transformation mapping $(I,I_A,I_B,I_C)$ to $(G,G_A,G_B,G_C)$ and under this transformation, the six-point circle will go to some conic through the 6 points which you want.