HMMT 二月 2025 · 几何 · 第 8 题

HMMT February 2025 — Geometry — Problem 8

Let ABCD be an isosceles trapezoid such that CD > AB = 4. Let E be a point on line CD such that DE = 2 and D lies between E and C . Let M be the midpoint of AE . Given that points A , B , C , D , and M lie on a circle with radius 5, compute M D . ◦

解析

Let ABCD be an isosceles trapezoid such that CD > AB = 4. Let E be a point on line CD such that DE = 2 and D lies between E and C . Let M be the midpoint of AE . Given that points A , B , C , D , and M lie on a circle with radius 5, compute M D . Proposed by: Sarunyu Thongjarast √ Answer: 6 Solution: A B ′ D M E C D ′ ′ ′ Let D be the reflection of D across M . Then, ADED is a parallelogram. Hence, D A = 2, so √ ′ ′ ′ D B = 6. Thus, if D M = M D = x , then Power of a Point at D gives x · (2 x ) = 2 · 6, so x = 6 . Remark. The radius of the circle is unnecessary. ◦