SMO 2013 Problem 2

Let $ABC$ be an acute-angled triangle and let $D$ , $E$ , and $F$ be the midpoints of $BC$ , $CA$ , and $AB$ respectively. Construct a circle, centered at the orthocenter of triangle $ABC$ , such that triangle $ABC$ lies in the interior of the circle. Extend $EF$ to intersect the circle at $P$ , $FD$ to intersect the circle at $Q$ and $DE$ to intersect the circle at $R$ . Show that $AP=BQ=CR$ .