HMMT February, 2025

Geometry Difficulty 4.8 AIME Prove it United States

Problem:
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 $\overline{AE}$ . Given that points $A$ , $B$ , $C$ , $D$ , and $M$ lie on a circle with radius $5$ , compute $MD$ .

Solution

Solution:

Let $D'$ be the reflection of $D$ across $M$ . Then, $ADED'$ is a parallelogram. Hence, $D'A = 2$ , so $D'B = 6$ . Thus, if $D'M = MD = x$ , then Power of a Point at $D'$ gives $x \cdot (2x) = 2 \cdot 6$ , so $x = \sqrt{6}$ .

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