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9 HS^2 + 4 (AH * AI + BH * BI + CH * CI) >= 3d^2

Source: 6th German TST 2005, problem 3

May 30, 2005
geometryincentercircumcircleinequalitiesinradiusEulervector

Problem Statement

Let ABCABC be a triangle with orthocenter HH, incenter II and centroid SS, and let dd be the diameter of the circumcircle of triangle ABCABC. Prove the inequality 9⋅HS2+4(AH⋅AI+BH⋅BI+CH⋅CI)≥3d2,9\cdot HS^2+4\left(AH\cdot AI+BH\cdot BI+CH\cdot CI\right)\geq 3d^2, and determine when equality holds.
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