Problem

Source: Romania TST 2 2012, Problem 2

Tags: geometry, circumcircle, trigonometry, symmetry, geometric transformation, reflection, rectangle



Let $ABCD$ be a convex circumscribed quadrilateral such that $\angle ABC+\angle ADC<180^{\circ}$ and $\angle ABD+\angle ACB=\angle ACD+\angle ADB$. Prove that one of the diagonals of quadrilateral $ABCD$ passes through the other diagonals midpoint.