Problem of the Day #97

POTD March 15, 2024

In parallelogram ABCD, E is a point on the ray \vec{AD}. It is given that BE intersects lines AC, CD at F,G respectively, BF = EG, and BC = 3. Then, the length of AE can be written as \frac{a+b\sqrt c}{d}, where a,b,c, and d,are positive integers with c not divisible by the square of any prime and \gcd(a,b,d)=1. Find a+b+c+d.