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Cyclic with Incenter and Orthocenter

Source: 2020 China North Mathematical Olympiad Basic Level P4

August 4, 2020
geometryincenterorthocentergeometry solvedCircumcentercyclic quadrilateralSpiral Similarity

Problem Statement

In △ABC\triangle ABC, ∠BAC=60∘\angle BAC = 60^{\circ}, point DD lies on side BCBC, O1O_1 and O2O_2 are the centers of the circumcircles of △ABD\triangle ABD and △ACD\triangle ACD, respectively. Lines BO1BO_1 and CO2CO_2 intersect at point PP. If II is the incenter of △ABC\triangle ABC and HH is the orthocenter of △PBC\triangle PBC, then prove that the four points B,C,I,HB,C,I,H are on the same circle.
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