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Perpendicularity from Midpoints

Source: 2011 Philippine Math Olympiad National Stage Problem 2

July 15, 2013
geometrygeometric transformationreflection

Problem Statement

In triangle ABCABC, let XX and YY be the midpoints of ABAB and ACAC, respectively. On segment BCBC, there is a point DD, different from its midpoint, such that ∠XDY=∠BAC\angle{XDY}=\angle{BAC}. Prove that AD⊥BCAD\perp BC.
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