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The perpendicular bisector of the bisector

Source: Iran TST 2013:TST 3,Day 2,Problem 2

July 26, 2013
Asymptotegeometrycircumcircleangle bisectorperpendicular bisectorgeometry proposed

Problem Statement

In triangle ABCABC, ADAD and AHAH are the angle bisector and the altitude of vertex AA, respectively. The perpendicular bisector of ADAD, intersects the semicircles with diameters ABAB and ACAC which are drawn outside triangle ABCABC in XX and YY, respectively. Prove that the quadrilateral XYDHXYDH is concyclic.
Proposed by Mahan Malihi