- Most likely we have seen this before, since there were here

several discusions on concurrent circles.

Anyway, here is it:

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

Denote

A' := (reflection of BP in BQ) /\ (reflection of CP in CQ)

B' := (reflection of P in CQ) /\ (reflection of AP in AQ)

C' := (reflection of AP in AQ) /\ (reflection of BP in BQ)

The Circles A'BC, B'CA, C'AB are concurrent at some point R.

Question:

Which are the coordinates of R in terms of the

coordinates of P and Q.

Special Case: P, Q = P* : Isogonal conjugate points.

APH - The circumcircles of AaAbAc, BaBbBc, CaCbCc are concurrent on the NPC circle of A'B'C'.AaAbAc, BaBbBc, CaCbCc = the pedal triangles of A", B", C", wrt triangle A'B'C' resp.Let ABC be a triangle, P a point andÂ A'B'C' the pedal triangle of P.Denote:L = the Euler line of A'B'C'A", B", C" = orthogonal projections of A, B, C, on L, resp.The point of concurrence is the reflection point of L wrt the medial triangle of A'B'C'.Problem:Let ABC be a triangle and L a line.For which points P's the L is the Euler line of the pedal triangle of P ?APH