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circle ext. tangent to 3 circles, 13-14-15 triangle related

Source: 2019 Dürer Math Competition Finals Day2 E+16 https://artofproblemsolving.com/community/c1621835_2019_

January 5, 2022
geometrytangent circles

Problem Statement

Triangle ABCABC has side lengths 1313, 1414 and 1515. Let k,kA,kB,kCk, k_A,k_B,k_C be four circles of radius r r inside the triangle such that kAk_A is tangent to sides ABAB and ACAC, kBk_B is tangent to sides BABA and BCBC, kCk_C is tangent to sides CACA and CBCB, and kk is externally tangent to circles kAk_A, kBk_B and kCk_C. Let r=m/nr = m/n where mm and nn are coprime. Find m+nm + n.
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