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Inequality with circumcircle of foots in a triangle

Source: CMO 2023 P3

March 11, 2023
geometrycircumcircleInequalityinequalitiesCMOCMO 2023

Problem Statement

An acute triangle is a triangle that has all angles less than 90∘90^{\circ} (90∘90^{\circ} is a Right Angle). Let ABCABC be an acute triangle with altitudes ADAD, BEBE, and CFCF meeting at HH. The circle passing through points DD, EE, and FF meets ADAD, BEBE, and CFCF again at XX, YY, and ZZ respectively. Prove the following inequality: AHDX+BHEY+CHFZ≥3.\frac{AH}{DX}+\frac{BH}{EY}+\frac{CH}{FZ} \geq 3.
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