HYACINTHOS 23265

[Antreas P. Hatzipolakis]:

Sorry if I have posted this before.....

Let ABC be a triangle and P,Q two points.

Denote:

Pa = the reflection of P in QA
Qa = the reflection of Q in PA
Ma = the midpoint of PaQa.
Similarly Mb, Mc.

If Q is fixed, which is the locus of P such that

ABC, MaMbMc are perspective?

Special case:

P,Q = isogonal conjugate points

(for P,Q = O, H, the ABC, MaMbMc are homothetic)

APH

Locus for perspective and homothecy = {McCay cubic K003} U Ucircles{B,C,B’,C’} U Ucircles{B,C,A’}, where A’,B’,C’ are the excenters of ABC

 

For P=O, the homothetic center is X(49)

 

Regards

César Lozada