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Reflection across midpoint

Source: Moldova TST 2013

April 17, 2013
geometrygeometric transformationreflectioncircumcircleratioLaTeXangle bisector

Problem Statement

Consider the triangle △ABC\triangle ABC with AB≠ACAB \not = AC. Let point OO be the circumcenter of △ABC\triangle ABC. Let the angle bisector of ∠BAC\angle BAC intersect BCBC at point DD. Let EE be the reflection of point DD across the midpoint of the segment BCBC. The lines perpendicular to BCBC in points D,ED,E intersect the lines AO,ADAO,AD at the points X,YX,Y respectively. Prove that the quadrilateral B,X,C,YB,X,C,Y is cyclic.
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