Euler Line

Let ABC be a triangle, Aa,Bb,Cc the orth. projections of A,B,C on the Euler line, resp., A1B1C1, A2B2C2 the medial, orthic triangles, resp. and P a point.

Let A'B'C', A"B"C" be the circumcevian triangles of P with respect the triangles A1B1C1, A2B2C2, resp.

Which is the locus of P such that the lines:

1. A'Aa, B'Bb, C'Cc

2. A"Aa, B"Bb, C"Cc

are concurrent?

The Euler line + ??

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Generalization:

P,P* = two isogonal conjugate points.

A1B1C1, A2B2C2 = the pedal triangles of P,P*, resp.

Aa, Bb, Cc = the orth. projections of A,B,C on PP*, resp.

A'B'C', A"B"C" = the circumcevian triangles of a point Q with respect A1B1C1, A2B2C2, resp.

Which is the locus of Q such that the lines:

1. A'Aa, B'Bb, C'Cc

2. A"Aa, B"Bb, C"Cc

are concurrent?

Is it the line PP* + ??

Locus of point of concurrence? Common Circumcircle of A1B1C1 and A2B2C2 + ??

APH, 13 February 2012