MEMO 2013 Problem 5

Let $ABC$ be and acute triangle. Construct a triangle $PQR$ such that $AB = 2PQ$ , $BC = 2QR$ , $CA = 2 RP$ , 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.)