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right triangle exradii criterions

Source: 1986 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p2

November 21, 2022
geometryexcircleexradiusright triangle

Problem Statement

A triangle has sides a,b,ca, b,c, radius of the incircle rr and radii of the excircles ra,rb,rcr_a, r_b, r_c: Prove that: a) The triangle is right-angled if and only if: r+ra+rb+rc=a+b+cr + r_a + r_b + r_c = a + b + c. b) The triangle is right-angled if and only if: r2+ra2+rb2+rc2=a2+b2+c2r^2 + r^2_a + r^2_b + r^2_c = a^2 + b^2 + c^2.
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