2022 SSMO Team Round Problems/Problem 3
Problem
Let 
be an isosceles trapezoid such that 
Let 
be a point on 
such that 
Let the midpoint of 
be 
such that 
intersects 
at 
and 
at 
If 
and 
then ![$[AGF]$](http://latex.artofproblemsolving.com/8/a/d/8adc7350215cb64aa533d094e40d69b4c4e1fc6c.png)
can be expressed as 
where 
and 
are relatively prime positive integers. Find 
![[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]](http://latex.artofproblemsolving.com/c/0/2/c028545fd593c232a4e030446bbfee76ac0fb772.png)
