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in-radius, ex-radii

Source: 3rd German pre-TST 2005, 12 Feb 2005, Problem 3

February 24, 2005
geometryinradiustrigonometryinequalitiesfunctiontriangle inequalitygeometry solved

Problem Statement

Let ABC be a triangle and let r,ra,rb,rcr, r_a, r_b, r_c denote the inradius and ex-radii opposite to the vertices A,B,CA, B, C, respectively. Suppose that a>ra,b>rb,c>rca>r_a, b>r_b, c>r_c. Prove that (a) △ABC\triangle ABC is acute. (b) a+b+c>r+ra+rb+rca+b+c > r+r_a+r_b+r_c.
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