I believe I have a synthetic proof of the theorem you mention. Is the MT point associated to P the complement of the isotomic conjugate of P? This has been called the "isotom-complement" by Grinberg.
If the answer to this question is yes, then I have such a proof, which I found several years ago with my student Igor Minevich.
--- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...> wrote:
> ---------- Forwarded message ----------
> From: Francisco Bellot Rosado <franciscobellot@...>
> Date: Wed, Oct 24, 2012 at 11:08 AM
> Subject: About a problem quoted in Hyacinthus
> To: anopolis72@...
> Dear Prof. Hatzipolakis,
> my name is Francisco Bellot and I am very interested in a very nice
> problem, partly quoted in the message of Hyacinthus #18776, dated
> March 30, 2010:
> "When points are cyclocevian conjugates, their MT (Miquel associated
> points) are isogonal conjugates".
> I know a proof with barycentric coordinates, but I wonder if there is
> also a synthetic proof and, moreover, the origin of this problem. I
> think it would be known by the french geometers of the XIX Century,
> but my search on it (FGM, Mathesis, etc) have been unsuccesful. I
> would be very grateful to you if you can help me on this problem.
> Best regards,
> Francisco Bellot
> Europe representative of the World Federation of National Maths Competitions
> Editor, Digital journal REOIM (Revista Escolar de la Olimpiada
> Iberoamericana de Matemática)