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Clean geometric inequality

Source: pOMA 2024/3

November 14, 2024
geometrycircumcirclegeometric inequalityinequalities

Problem Statement

Let ABCABC be a triangle with circumcircle Ω\Omega, and let PP be a point on the arc BCBC of Ω\Omega not containing AA. Let ωB\omega_B and ωC\omega_C be circles respectively passing through BB and CC and such that both of them are tangent to line APAP at point PP. Let RR, RBR_B, RCR_C be the radii of Ω\Omega, ωB\omega_B, and ωC\omega_C, respectively. Prove that if hh is the distance from AA to line BCBC, then RB+RCR≤BCh. \frac{R_B+R_C}{R} \le \frac{BC}{h}.
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