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Balkan MO 2014 shortlist G-7

Source: Balkan MO 2014 shortlist G-7

June 12, 2015
geometryincenterorthocenter

Problem Statement

Let II be the incenter of △ABC\triangle ABC and let HaH_a, HbH_b, and HcH_c be the orthocenters of △BIC\triangle BIC , △CIA\triangle CIA, and △AIB\triangle AIB, respectively. The lines HaHbH_aH_b meets ABAB at XX and the line HaHcH_aH_c meets ACAC at YY. If the midpoint TT of the median AMAM of △ABC\triangle ABC lies on XYXY, prove that the line HaTH_aT is perpendicular to BCBC
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