2002 USAMO 第 2 题

Let $ABC$ be a triangle such that

$$\left( \cot \frac{A}{2} \right)^2 + \left( 2 \cot \frac{B}{2} \right)^2 + \left( 3 \cot \frac{C}{2} \right)^2 = \left( \frac{6s}{7r} \right)^2,$$

where $s$ and $r$ denote its semiperimeter and inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisor and determine those integers.

令 $ABC$ 为三角形，使得

$$\left( \cot \frac{A}{2} \right)^2 + \left( 2 \cot \frac{B}{2} \right)^2 + \left( 3 \cot \frac{C}{2} \right)^2 = \left( \frac{6s}{7r} \right)^2,$$

其中$s$和$r$分别表示其半周长和内半径。证明三角形 $ABC$ 与边长均为无公约数的正整数的三角形 $T$ 相似，并确定这些整数。