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QO_|_AC if QP _|_ KM, QM//BO - All-Russian MO 1999 Regional (R4) 9.8

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September 25, 2024
geometryperpendicular

Problem Statement

In triangle ABCABC (AB>BCAB > BC), KK and MM are the midpoints of sides ABAB and ACAC, OO is the point of intersection of the angle bisectors. Let PP be the intersection point of lines KMKM and COCO, and the point QQ is such that QP⊥KMQP \perp KM and QM∥BOQM \parallel BO. Prove that QO⊥ACQO \perp AC.
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