HMMT 二月 2021 · 几何 · 第 6 题

HMMT February 2021 — Geometry — Problem 6

In triangle ABC , let M be the midpoint of BC , H be the orthocenter, and O be the circumcenter. 2 Let N be the reflection of M over H . Suppose that OA = ON = 11 and OH = 7. Compute BC .

解析

In triangle ABC , let M be the midpoint of BC , H be the orthocenter, and O be the circumcenter. 2 Let N be the reflection of M over H . Suppose that OA = ON = 11 and OH = 7. Compute BC . Proposed by: Milan Haiman Answer: 288 Solution: Let ω be the circumcircle of 4 ABC . Note that because ON = OA , N is on ω . Let P be the reflection of H over M . Then, P is also on ω . If Q is the midpoint of N P , note that because N H = HM = M P, Q is also the midpoint of HM . Since OQ ⊥ N P , we know that OQ ⊥ HM . As Q is also the midpoint of HM , OM = OH = 7 . With this, √ √ 2 2 BM = OB − BM = 6 2 , √ 2 and BC = 2 BM = 12 2. Therefore, BC = 288.