Perspective

Let ABC be a triangle and OaObOc,HaHbHc the pedal triangles of O,H, resp.

Denote

A' = HaN /\ AO

B' = HbN /\ BO

C' = HcN /\ CO

Are the triangles OaObOc, A'B'C' perspective?

Generalization:

Let ABC be a triangle, P,P* two isogonal conjugate points, PaPbPc, P1P2P3 the pedal triangles of P, P*, resp. and M the midpoint of PP* (= the circumcenter of the common pedal circle of P amd P*).

Denote:

A' = P1M /\ AP

B' = P2M /\ BP

C' = P3M /\ CP

Which is the locus of P such that PaPbPc, A'B'C' are perspective?

APH, 13 January 2012

-------------------------

The locus is a septic through I and H, and a nonic through I and O.

For P=I, H, O, the respective perpectors are I, H and X216.

Francisco Javier García Capitán
13 January 2012