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locus of incenters of A_1B_1C_1 when <B_1A_1C_1 + 2 <BAC = 180^o ...

Source: VII Soros Olympiad 2000-01 Round 1 , 10.5

July 4, 2021
geometryincenterLocusangles

Problem Statement

An acute-angled triangle ABCABC is given. Points A1,B1A_1, B_1 and C1C_1 are taken on its sides BC,CABC, CA and ABAB, respectively, such that ∠B1A1C1+2∠BAC=180o\angle B_1A_1C_1 + 2 \angle BAC = 180^o, ∠A1C1B1+2∠ACB=180o\angle A_1C_1B_1 + 2 \angle ACB = 180^o, ∠C1B1A1+2∠CBA=180o\angle C_1B_1A_1 + 2 \angle CBA = 180^o. Find the locus of the centers of the circles inscribed in triangles A1B1C1A_1B_1C_1 (all kinds of such triangles are considered).
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