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ABC isosceles if MN // PQ, incircle (Kyiv City Olympiad 2006 10.4)

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June 28, 2020
geometryisoscelesparallelincircle

Problem Statement

A circle ω\omega is inscribed in the acute-angled triangle △ABC\vartriangle ABC, which touches the side BCBC at the point KK. On the lines ABAB and ACAC, the points PP and QQ, respectively, are chosen so that PK⊥ACPK \perp AC and QK⊥ABQK \perp AB. Denote by MM and NN the points of intersection of KPKP and KQKQ with the circle ω\omega. Prove that if MN∥PQMN \parallel PQ, then △ABC\vartriangle ABC is isosceles.
(S. Slobodyanyuk)
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