RO-NMO 2006 Problem 73

In the acute-angle triangle $ABC$ we have $\angle ACB = 45^\circ$ . The points $A_1$ and $B_1$ are the feet of the altitudes from $A$ and $B$ , and $H$ is the orthocenter of the triangle. We consider the points $D$ and $E$ on the segments $AA_1$ and $BC$ such that $A_1D = A_1E = A_1B_1$ . Prove that

a) $A_1B_1 = \sqrt{ \frac{A_1B^2+A_1C^2}{2} }$ ;

b) $CH=DE$ .