NL-NWO 2022 Problem 4

In triangle $ABC$ , the point $D$ lies on segment $AB$ such that $CD$ is the angle bisector of angle $\angle C$ . The perpendicular bisector of segment $CD$ intersects the line $AB$ in $E$ . Suppose that $|BE| = 4$ and $|AB| = 5$ . (a) Prove that $\angle BAC = \angle BCE$ . (b) Prove that $2|AD| = |ED|$ .