USAMO 2007 Problem 6

Let $ABC$ be an acute triangle with $\omega,S$ , and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_{A}$ is tangent internally to $S$ at $A$ and tangent externally to $\omega$ . Circle $S_{A}$ is tangent internally to $S$ at $A$ and tangent internally to $\omega$ . Let $P_{A}$ and $Q_{A}$ denote the centers of $\omega_{A}$ and $S_{A}$ , respectively. Define points $P_{B}, Q_{B}, P_{C}, Q_{C}$ analogously. Prove that $$8P_{A}Q_{A}\cdot P_{B}Q_{B}\cdot P_{C}Q_{C}\leq R^{3} ,$$ with equality if and only if triangle $ABC$ is equilateral.