Let ABC be a triangle, Q a point, QaQbQc the pedal triangle of Q, (X) the circumcircle of QaQbQc (=pedal circle of Q), P a point on the OQ line and PaPbPc the pedal triangle of P.
(X1), (X2), (X3) the reflections of (X) in OPa,OPb,OPc, resp.
A'B'C' = the triangle bounded by the radical axes of ((O),(X1)),((O),(X2)),((O),(X3))
The triangles A'B'C', X1X2X3 are perspective at a point Qp.
Antreas P. Hatzipolakis, 13 Feb. 2013