Middle European Mathematical Olympiad 2013 Problem T-5

Let $ABC$ be an acute triangle. Construct a triangle $PQR$ such that $AB = 2PQ$, $BC = 2QR$, $CA = 2RP$, and the lines $PQ$, $QR$, and $RP$ pass through the points $A$, $B$, and $C$, respectively. (All six points $A$, $B$, $C$, $P$, $Q$, and $R$ are distinct.)