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R/2r >= 1/ sin a/2 (1- sin a/2)

Source: Polish MO Finals 1975 p5

August 23, 2024
geometryGeometric Inequalitiestrigonometry

Problem Statement

Show that it is possible to circumscribe a circle of radius RR about, and inscribe a circle of radius rr in some triangle with one angle equal to aa, if and only if 2Rr≥1sin⁡a2(1−sin⁡a2)\frac{2R}{r} \ge \dfrac{1}{ \sin \frac{a}{2} \left(1- \sin \frac{a}{2} \right)}
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