AT-OEMO-REG 2010 Problem 3

Let $\triangle ABC$ be a triangle and let $D$ be a point on side $\overline{BC}$ . Let $U$ and $V$ be the circumcenters of triangles $\triangle ABD$ and $\triangle ADC$ , respectively. Show, that $\triangle ABC$ and $\triangle AUV$ are similar.

(41th Austrian Mathematical Olympiad, regional competition, problem 3)