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collinearity of three circumcenters and one incenter

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October 31, 2019
geometrycircumcircleinradiusincenter

Problem Statement

Let O O be the circumcenter of a triangle ABC ABC with ∠BAC=60∘ \angle BAC=60^{\circ } whose incenter is denoted by I. I. Let B1,C1 B_1,C_1 be the intersection of BI,CI BI,CI with the circumcircle of ABC, ABC, respectively. Denote by O1,O2 O_1,O_2 the circumcenters of BIC1,CIB1, BIC_1,CIB_1, respectively. Show that O1,I,O,O2 O_1,I,O,O_2 are collinear.
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