ORTHOLOGIC TRIANGLES - LOCUS

1.

Let ABC be a triangle and A'B'C' the pedal triangle of O (= medial triangle).

Denote:

A" = (O, OA')/\((A'B'C') - A') = the other than A' intersection of the circle centered at O with radius OA' and the pedal circle of O(=NPC)

B" = (O, OB')/\((A'B'C') - B')

C" = (O, OC')/\((A'B'C') - C')

The triangles ABC, A"B"C" are orthologic (with orthologic centers O1,O2 on NPC and circumcircle)

Locus:

Let ABC be a triangle, A'B'C' the pedal triangle of point P and A",B",C" the second intersections of the circles (P,PA'), (P,PB'), (P,PC') with the pedal circle of P, resp.

Which is the locus of P such that the triangles ABC, A"B"C" are orthologic?

2.

Let ABC be a triangle, A'B'C' the pedal triangle of H (=orthic triangle) and A*B*C* the circumcevian triangle of H wrt A'B'C'.

Denote:

A" = (H, HA*) /\ (A*B*C* - A*) = the other intersection of the circle centered at H with radius HA* and the circle (A*B*C*) = (A'B'C') = pedal circle of H.

B" = (H, HB*) /\ (A*B*C* - B*)

C" = (H, HC*) /\ (A*B*C* - C*)

The triangles ABC, A"B"C" are orthologic (with orthologic centers H1,H2 on NPC and circumcircle)

Locus:

Let ABC be a triangle, A'B'C' the pedal triangle of point P, A*B*C* the circumcevian triangle of P wrt A'B'C' and A",B",C" the second intersections of the circles (P,PA*), (P,PB*), (P,PC*) with the pedal circle of P, resp.

Which is the locus of P such that the triangles ABC, A"B"C" are orthologic?

Antreas P. Hatzipolakis, 15 March 2013