1997 APMO 第 1 题

Let $ABC$ be a triangle inscribed in a circle and let

$$l_a = \frac{m_a}{M_a} \ , \ \ l_b = \frac{m_b}{M_b} \ , \ \ l_c = \frac{m_c}{M_c} \ ,$$

where $m_a$ , $m_b$ , $m_c$ are the lengths of the angle bisectors (internal to the triangle) and $M_a$ , $M_b$ , $M_c$ are the lengths of the angle bisectors extended until they meet the circle. Prove that

$$\frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \geq 3$$

and that equality holds iff $ABC$ is an equilateral triangle.

令 $ABC$ 为一个内切于圆的三角形，并令

$$l_a = \frac{m_a}{M_a} \ , \ \ l_b = \frac{m_b}{M_b} \ , \ \ l_c = \frac{m_c}{M_c} \ ,$$

其中 $m_a$ 、 $m_b$ 、 $m_c$ 是角平分线(三角形内部)的长度， $M_a$ 、 $M_b$ 、 $M_c$ 是角平分线延伸直到与圆相交的角平分线的长度。证明

$$\frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \geq 3$$

并且当且仅当 $ABC$ 是等边三角形时，该等式成立。