AT-OEMO-BEG 2010 Problem 4

In the right-angled triangle $ABC$ with a right angle at $C$ , the side $BC$ is longer than the side $AC$ . The perpendicular bisector of $AB$ intersects the line $BC$ at point $D$ and the line $AC$ at point $E$ . The segments $DE$ has the same length as the side $AB$ . Find the measures of the angles of the triangle $ABC$ .

(R. Henner, Vienna)