DE-BUNDESWETTBEWERB 2007 Problem 3

In triangle $ABC$ points $E$ and $F$ lie on sides $AC$ and $BC$ such that segments $AE$ and $BF$ have equal length, and circles formed by $A,C,F$ and by $B,C,E,$ respectively, intersect at point $C$ and another point $D.$ Prove that that the line $CD$ bisects $\angle ACB.$