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intersection of two lines lies on the circumcircle

Source: KJMO 2013 p5

May 1, 2019
geometrycircumcircleacuteCircumcenterperpendicularmidpoint

Problem Statement

In an acute triangle △ABC,∠A>∠B\triangle ABC, \angle A > \angle B. Let the midpoint of ABAB be DD, and let the foot of the perpendicular from AA to BCBC be EE, and BB from CACA be FF. Let the circumcenter of △DEF\triangle DEF be OO. A point JJ on segment BEBE satisfi es ∠ODC=∠EAJ\angle ODC = \angle EAJ. Prove that AJ∩DCAJ \cap DC lies on the circumcircle of △BDE\triangle BDE.
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