GULF 2016 Problem 3

Consider the acute-angled triangle $ABC$ . Let $X$ be a point on the side $BC$ , and $Y$ be a point on the side $CA$ . The circle $k_1$ with diameter $AX$ cuts $AC$ again at $E'$ .The circle $k_2$ with diameter $BY$ cuts $BC$ again at $B'$ . (i) Let $M$ be the midpoint of $XY$ . Prove that $A'M = B'M$ . (ii) Suppose that $k_1$ and $k_2$ meet at $P$ and $Q$ . Prove that the orthocentre of $ABC$ lies on the line $PQ$ .