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Harmonic quadrilateral(s) and a bit of nine-point circle

Source: KoMaL A. 873

March 12, 2024
geometryHarmonic Quadrilateralprojective geometrykomal

Problem Statement

Let ABCDABCD be a convex cyclic quadrilateral satisfying AB⋅CD=AD⋅BCAB\cdot CD=AD\cdot BC. Let the inscribed circle ω\omega of triangle ABCABC be tangent to sides BCBC, CACA and ABAB at points A′,B′A', B' and C′C', respectively. Let point KK be the intersection of line IDID and the nine-point circle of triangle A′B′C′A'B'C' that is inside line segment IDID. Let SS denote the centroid of triangle A′B′C′A'B'C'. Prove that lines SKSK and BB′BB' intersect each other on circle ω\omega.
Proposed by Áron Bán-Szabó, Budapest
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