HMMT 二月 2003 · 冲刺赛 · 第 25 题

英文解析

25. Let ABC be an isosceles triangle with apex A . Let I be the incenter. If AI = 3 andthe distance from I to BC is 2, then what is the length of BC ?
    √
    Solution: 4 5
    Let X and Y be the points where the incircle touches AB and BC , respectively.
    Then AXI and AY B are similar right triangles. Since I is the incenter, we have
    √
    IX = IY = 2. Using the Pythagorean theorem on triangle AXI , we find AX = 5.
    √
    By similarity, AY /AX = BY /IX . Plugging in the numbers given, 5 / 5 = BY / 2, so
    √ √
    BY = 2 5. Y is the midpoint of BC , so BC = 4 5. 2