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Points with equal angles

Source: 2015 USAJMO problem 5

April 29, 2015
USA(J)MOUSAJMOgeometrycyclic quadrilateral2015 USAJMOsimilar trianglestrigonometry

Problem Statement

Let ABCDABCD be a cyclic quadrilateral. Prove that there exists a point XX on segment BD‾\overline{BD} such that ∠BAC=∠XAD\angle BAC=\angle XAD and ∠BCA=∠XCD\angle BCA=\angle XCD if and only if there exists a point YY on segment AC‾\overline{AC} such that ∠CBD=∠YBA\angle CBD=\angle YBA and ∠CDB=∠YDA\angle CDB=\angle YDA.
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