2022 SSMO Team Round Problems/Problem 3

Problem

Let $ABCD$ be an isosceles trapezoid such that $AB\parallel CD.$ Let $E$ be a point on $CD$ such that $AB=CE.$ Let the midpoint of $DE$ be $M$ such that $BD$ intersects $AM$ at $G$ and $AE$ at $F.$ If $DC=36, AB=24,$ and $AD=10,$ then $[AGF]$ can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

$[asy] unitsize(2mm); fill((6,8/5)--(10,8/3)--(6,8)--cycle,lightgray); draw((0,0)--(36,0)--(30,8)--(6,8)--cycle); draw((6,8)--(12,0)--(0,0)); draw((0,0)--(30,8)); draw((6,8)--(6,0)); label("A", (6,8), NW); dot((6,8)); label("B", (30,8), NE); dot((30,8)); label("C", (36,0), SE); dot((36,0)); label("D", (0,0), SW); dot((0,0)); label("E",(12,0),S); dot((12,0)); label("M",(6,0),S); dot((6,0)); dot((6,8/5)); dot((10,8/3)); [/asy]$

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2022 SSMO Team Round Problems/Problem 3

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