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Circumtriangular Hyperbola

Source: ISL 2022/G6

July 9, 2023
conicshyperbolageometryprojective geometry

Problem Statement

Let ABCABC be an acute triangle with altitude AH‾\overline{AH}, and let PP be a variable point such that the angle bisectors kk and ℓ\ell of ∠PBC\angle PBC and ∠PCB\angle PCB, respectively, meet on AH‾\overline{AH}. Let kk meet AC‾\overline{AC} at EE, ℓ\ell meet AB‾\overline{AB} at FF, and EF‾\overline{EF} meet AH‾\overline{AH} at QQ. Prove that as PP varies, line PQPQ passes through a fixed point.
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