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Orthocenter of intriangle farther than incenter

Source: 2016 IMO Shortlist G8

July 19, 2017
geometryincenterIMO Shortlistorthocenter

Problem Statement

Let A1,B1A_1, B_1 and C1C_1 be points on sides BCBC, CACA and ABAB of an acute triangle ABCABC respectively, such that AA1AA_1, BB1BB_1 and CC1CC_1 are the internal angle bisectors of triangle ABCABC. Let II be the incentre of triangle ABCABC, and HH be the orthocentre of triangle A1B1C1A_1B_1C_1. Show that AH+BH+CH≥AI+BI+CI.AH + BH + CH \geq AI + BI + CI.
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