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Geometry Mathley 10.4 2n incenters collinear

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June 7, 2020
geometryincenterbicentric

Problem Statement

Let A1A2A3...AnA_1A_2A_3...A_n be a bicentric polygon with nn sides. Denote by IiI_i the incenter of triangle Ai−1AiAi+1,Ai(i+1)A_{i-1}A_iA_{i+1}, A_{i(i+1)} the intersection of AiAi+2A_iA_{i+2} and Ai−1Ai+1,Ii(i+1)A_{i-1}A_{i+1},I_{i(i+1)} is the incenter of triangle AiAi(i+1)Ai+1A_iA_{i(i+1)}A_{i+1} (i=1,ni = 1, n). Prove that there exist 2n2n points I1,I2,...,In,I12,I23,....,In1I_1, I_2, ..., I_n, I_{12}, I_{23}, ...., I_{n1} on the same circle.
Nguyễn Văn Linh
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