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Perpendicular to line through orthocenter and incenter

Source: CGMO 2020 Day2 P7

August 10, 2020
geometrycircumcircleincenter

Problem Statement

Let OO be the circumcenter of triangle △ABC\triangle ABC, where ∠BAC=120∘\angle BAC=120^{\circ}. The tangent at AA to (ABC)(ABC) meets the tangents at B,CB,C at (ABC)(ABC) at points P,QP,Q respectively. Let H,IH,I be the orthocenter and incenter of △OPQ\triangle OPQ respectively. Define M,NM,N as the midpoints of arc \overarc{BAC} and OIOI respectively, and let MNMN meet (ABC)(ABC) again at DD. Prove that ADAD is perpendicular to HIHI.
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