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Inscribed semicircles

Source: South Africa 2002

October 13, 2005
geometrytrigonometryincentergeometric transformationhomothetyratiosearch

Problem Statement

In acute-angled triangle ABCABC, a semicircle with radius rar_a is constructed with its base on BCBC and tangent to the other two sides. rbr_b and rcr_c are defined similarly. rr is the radius of the incircle of ABCABC. Show that 2r=1ra+1rb+1rc. \frac{2}{r} = \frac{1}{r_a} + \frac{1}{r_b} + \frac{1}{r_c}.
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