ELMO SL 2014 G5

Let $P$ be a point in the interior of an acute triangle $ABC$ , and let $Q$ be its isogonal conjugate. Denote by $\omega_P$ and $\omega_Q$ the circumcircles of triangles $BPC$ and $BQC$ , respectively. Suppose the circle with diameter $\overline{AP}$ intersects $\omega_P$ again at $M$ , and line $AM$ intersects $\omega_P$ again at $X$ . Similarly, suppose the circle with diameter $\overline{AQ}$ intersects $\omega_Q$ again at $N$ , and line $AN$ intersects $\omega_Q$ again at $Y$ .

Prove that lines $MN$ and $XY$ are parallel.

(Here, the points $P$ and $Q$ are isogonal conjugates with respect to $\triangle ABC$ if the internal angle bisectors of $\angle BAC$ , $\angle CBA$ , and $\angle ACB$ also bisect the angles $\angle PAQ$ , $\angle PBQ$ , and $\angle PCQ$ , respectively. For example, the orthocenter is the isogonal conjugate of the circumcenter.)

Proposed by Sammy Luo