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midpoints wanted, 4 equal inradii in an equilateral triangle

Source: 1995 Bulgaria NMO, Round 4, p4

July 30, 2021
geometrymidpointequal circlesinradiiEquilateral

Problem Statement

Points A1,B1,C1A_1,B_1,C_1 are selected on the sides BCBC,CACA,ABAB respectively of an equilateral triangle ABCABC in such a way that the inradii of the triangles C1AB1C_1AB_1, A1BC1A_1BC_1, B1CA1B_1CA_1 and A1B1C1A_1B_1C_1 are equal. Prove that A1,B1,C1A_1,B_1,C_1 are the midpoints of the corresponding sides.
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