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 $2|AD| = |ED|$. [asy][asy] unitsize(1 cm); pair A, B, C, D, E; A = (0,0); B = (2,0); C = (1.8,1.8); D = extension(C, incenter(A,B,C), A, B); E = extension((C + D)/2, (C + D)/2 + rotate(90)*(C - D), A, B); draw((E + (0.5,0))--A--C--B); draw(C--D); draw(interp((C + D)/2,E,-0.3)--interp((C + D)/2,E,1.2)); dot("$A$", A, SW); dot("$B$", B, S); dot("$C$", C, N); dot("$D$", D, S); dot("$E$", E, S); [/asy][/asy]