COLLINEARITY

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

Denote:

PQa = the isogonal conjugate of P wrt QBC

PQb = the isogonal conjugate of P wrt QCA

PQc = the isogonal conjugate of P wrt QAB

QPa = the isogonal conjugate of Q wrt PBC

QPb = the isogonal conjugate of Q wrt PCA

QPc = the isogonal conjugate of Q wrt PAB

Apq = PQaQPa /\ BC

Bpq = PQbQPb /\ CA

Cpq = PQcQPc /\ AB

Conjecture: The points Apq, Bpq, Cpq are collinear.

Let R be another point. We have the lines (if the conjecture is true):

ApqBpqCpq, AqrBqrCqr, ArpBrpCrp.

Are they concurrent? Do they bound a triangle in perspective with ABC?

Antreas P. Hatzipolakis, 18 May 2013