MathNet 2016-part-02 — 184 problems (EN–VI) Question 175

Let $\triangle ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D$, $E$ and $F$ be points on the segments $BC$, $CA$ and $AB$ respectively such that $|BF| = |BE|$ and such that $ED$ is the interior bisector of $\angle BEC$.

Prove that $|BD| = |EF|$ if and only if $|AF| = |EC|$.