1993 IMO 第 2 题

(CAN 2) Let triangle $A B C$ be such that its circumradius $R$ is equal to 1. Let $r$ be the inradius of $A B C$ and let $p$ be the inradius of the orthic triangle $A^{\prime} B^{\prime} C^{\prime}$ of triangle $A B C$. Prove that $p \leq 1-\frac{1}{3}(1+r)^{2}$. Remark. The orthic triangle is the triangle whose vertices are the feet of the altitudes of $A B C$.

(CAN 2) 令三角形 $A B C$ 的外接圆半径 $R$ 等于 1。令 $r$ 为 $A B C$ 的内半径，令 $p$ 为三角形 $A B C$ 的正交三角形 $A^{\prime} B^{\prime} C^{\prime}$ 的内半径。证明 $p \leq 1-\frac{1}{3}(1+r)^{2}$。评论。正三角形是以 $A B C$ 的高脚为顶点的三角形。