MathNet 2013 Question 62

An isosceles triangle $ABE$ is inscribed in the square $ABCD$ of side $a$, with $E$ the midpoint of $CD$. Point $F$ is the foot of the altitude from $B$ to the leg $AE$ of triangle $ABE$. Prove that $|EF|:|FB|:|BE|=3:4:5$. Draw a suitable sketch.



Supplemental coordinate illustration (AI); perspective is not to scale.