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Ratio of incircle and excircle ? - JBMO Shortlist

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October 30, 2010
ratiogeometryincentergeometry proposed

Problem Statement

A triangle ABCABC is inscribed in the circle C(O,R)\mathcal{C}(O,R). Let α<1\alpha <1 be the ratio of the radii of the circles tangent to C\mathcal{C}, and both of the rays (AB(AB and (AC(AC. The numbers β<1\beta <1 and γ<1\gamma <1 are defined analogously. Prove that α+β+γ=1\alpha + \beta + \gamma =1.
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