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Geometric inequality with reflections and tangents

Source: 2023 Macedonian Team Selection Test P2

May 21, 2023
geometric inequalityinequalitiesgeometrygeometric transformationreflection

Problem Statement

Let ABCABC be an acute triangle such that AB<ACAB<AC and AB<BCAB<BC. Let PP be a point on the segment BCBC such that ∠APB=∠BAC\angle APB = \angle BAC. The tangent to the circumcircle of triangle ABCABC at AA meets the circumcircle of triangle APBAPB at Q≠AQ \neq A. Let Q′Q' be the reflection of QQ with respect to the midpoint of ABAB. The line PQPQ meets the segment AQ′AQ' at SS. Prove that 1AB+1AC>1CS.\frac{1}{AB}+\frac{1}{AC} > \frac{1}{CS}.
Proposed by Nikola Velov
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