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Circumscribed quadrilateral and equal sums of angles

Source: Romania TST 2 2012, Problem 2

May 10, 2012
geometrycircumcircletrigonometrysymmetrygeometric transformationreflectionrectangle

Problem Statement

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