2007 USAMO 第 6 题

Let $ABC$ be an acute triangle with $\omega$, $\Omega$, and $R$ being its incircle, circumcircle, and circumradius, respectively. The circle $\omega_A$ is tangent internally to $\Omega$ at $A$ and externally tangent to $\omega$. Circle $\Omega_A$ is internally tangent to $\Omega$ at $A$ and tangent internally to $\omega$. Let $P_A$ and $Q_A$ denote the centers of $\omega_A$ and $\Omega_A$, respectively. Define points $P_B$, $Q_B$, $P_C$, $Q_C$ analogously. Prove that

$$8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3,$$

with equality if and only if triangle $ABC$ is equilateral.

设 $ABC$ 为锐角三角形，$\omega$、$\Omega$ 和 $R$ 分别为其内切圆、外接圆和外接半径。圆 $\omega_A$ 在 $A$ 处与 $\Omega$ 内部相切，在 $A$ 处与 $\omega$ 外部相切。圆 $\Omega_A$ 在 $A$ 处与 $\Omega$ 内切，并与 $\omega$ 内切。令$P_A$和$Q_A$分别表示$\omega_A$和$\Omega_A$的中心。类似地定义点$P_B$、$Q_B$、$P_C$、$Q_C$。证明

$$8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3,$$

相等当且仅当三角形 $ABC$ 是等边的。