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XY passes through D for certain X, Y

Source: MEMO 2015, problem T-5

August 28, 2015
geometry

Problem Statement

Let ABCABC be an acute triangle with AB>ACAB>AC. Prove that there exists a point DD with the following property: whenever two distinct points XX and YY lie in the interior of ABCABC such that the points BB, CC, XX, and YY lie on a circle and ∠AXB−∠ACB=∠CYA−∠CBA\angle AXB-\angle ACB=\angle CYA-\angle CBA holds, the line XYXY passes through DD.
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