HMMT 二月 2003 · 冲刺赛 · 第 25 题

HMMT February 2003 — Guts Round — Problem 25

[9] Let ABC be an isosceles triangle with apex A . Let I be the incenter. If AI = 3 and the distance from I to BC is 2, then what is the length of BC ? 2

解析

Let ABC be an isosceles triangle with apex A . Let I be the incenter. If AI = 3 and the 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