IGO 2018 A1

Two circles $\omega_1,\omega_2$ intersect each other at points $A,B$ . Let $PQ$ be a common tangent line of these two circles with $P \in \omega_1$ and $Q \in \omega_2$ . An arbitrary point $X$ lies on $\omega_1$ . Line $AX$ intersects $\omega_2$ for the second time at $Y$ . Point $Y'\ne Y$ lies on $\omega_2$ such that $QY = QY'$ . Line $Y'B$ intersects $\omega_1$ for the second time at $X'$ . Prove that $PX = PX'$ .

Proposed by Morteza Saghafian