MathNet 2017-part-01 — 176 problems (EN–VI) Question 27

Let $ABCD$ be a convex cyclic quadrilateral with circumcenter $U$ whose diagonals are perpendicular to each other. Let $g$ be the line obtained by reflecting the diagonal $AC$ in the angle bisector of $\angle BAD$.
Prove that the point $U$ lies on the line $g$.