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inradius wanted, 3 equal inradii related

Source: II Soros Olympiad 1995-96 R2 10-11.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

June 3, 2024
geometryinradiusequal circles

Problem Statement

On sides BCBC, CACA and ABAB of triangle ABCABC, points A1A_1, B1B_1, C1C_1 are taken, respectively, so that the radii of the circles inscribed in triangles A1BC1A_1BC_1, AB1C1AB_1C_1 and A1B1CA_1B_1C are equal to each other and equal to rr. The radius of the circle inscribed in triangle A1B1C1A_1B_1C_1 is equal to r1r_1. Find the radius of the circle inscribed in triangle ABCABC.