US-TST 2002 Problem 5

Consider the family of nonisosceles triangles $ABC$ satisfying the property $AC^2 + BC^2 = 2 AB^2$ . Points $M$ and $D$ lie on side $AB$ such that $AM = BM$ and $\angle ACD = \angle BCD$ . Point $E$ is in the plane such that $D$ is the incenter of triangle $CEM$ . Prove that exactly one of the ratios $$\frac{CE}{EM}, \frac{EM}{MC}, \frac{MC}{CE}$$ is constant.