HYACINTHOS 24006

Antreas P. Hatzipolakis

 

Let ABC be a triangle and A'B'C' the pedal triangle of H.

Denote:

Ab, Ac = the reflections of A' in BO, CO, resp.

Bc, Ba = the reflections of B' in CO, AO, resp.

Ca, Cb = the reflections of C' in AO, BO, resp.

The NPCs of A'AbAc, B'BcBa, C'CaCb are concurrent.

Let Na, Nb, Nc be the NPC centers of A'AbAc, B'BcBa, C'CaCb, resp.

ABC, NaNbNc are orthologic.

 

APH


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GENERALIZATION:

Let ABC be a triangle, P, Q two isogonal conjugate points and A'B'C' the pedal triangle of Q.

Denote:

Ab, Ac = the reflections of A' in BP, CP, resp.

Bc, Ba = the reflections of B' in CP, AP, resp.

Ca, Cb = the reflections of C' in AP, BP, resp.

The NPCs of A'AbAc, B'BcBa, C'CaCb are concurrent.

Point of concurrence ?

Let Na, Nb, Nc be the NPC centers of A'AbAc, B'BcBa, C'CaCb, resp.

Which is the locus of P such that ABC, NaNbNc are orthologic ?

APH