$ABCD$ is a parallelogram. The point $E$ is in the segment $BC$, and $F$ is a point in the line $DC$ such that $AF$ pass through the point $E$ and intersect the segment $BD$ in the point $G$. If $AG=6$ and $GE=4$. How long is $EF$?
Triangle DGA ≈ Triangle BGE
GE/ GA = EB /AD
4/6 = EB / AD
2/3 = EB /AD { AD = BC }
2/3 = EB / BC
(2/3)BC = EB
So EC =(1/3) BC
So EB/ EC = 2/1
So EC /EB = 1/2
And
Triangle AEB ≈ FEC
EF / AE = EC /EB
EF /10 = 1/2
EF = 5