US-TST 2008 Problem 2

Let $P$ , $Q$ , and $R$ be the points on sides $BC$ , $CA$ , and $AB$ of an acute triangle $ABC$ such that triangle $PQR$ is equilateral and has minimal area among all such equilateral triangles. Prove that the perpendiculars from $A$ to line $QR$ , from $B$ to line $RP$ , and from $C$ to line $PQ$ are concurrent.