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equal segments from KJMO 2009

Source: KJMO 2009 p5

May 2, 2019
geometrycircumcircleperpendicularequal segments

Problem Statement

Acute triangle △ABC\triangle ABC satis es AB<ACAB < AC. Let the circumcircle of this triangle be OO, and the midpoint of BC,CA,ABBC,CA,AB be D,E,FD,E,F. Let PP be the intersection of the circle with ABAB as its diameter and line DFDF, which is in the same side of CC with respect to ABAB. Let QQ be the intersection of the circle with ACAC as its diameter and the line DEDE, which is in the same side of BB with respect to ACAC. Let PQ∩BC=RPQ \cap BC = R, and let the line passing through RR and perpendicular to BCBC meet AOAO at XX. Prove that AX=XRAX = XR.
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