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Excenters, incenters are concyclic

Source: Oral Moscow Geometry Olympiad 2024, 8-9.2

September 3, 2024
geometryincenter

Problem Statement

The bisector BLBL was drawn in the triangle ABCABC. Let the points I1I_1 and I2I_2 be centers of the circles inscribed in the triangles ABLABL and CBLCBL, and the points J1J_1 and J2J_2 be centers of the excircles of these triangles touching the side BLBL. Prove that the points I1I_1, I2I_2, J1J_1 and J2J_2 lie on the same circle.
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