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Projection inequality

Source: Own. Malaysian IMO TST 2022 P1

May 7, 2022
geometryinequalities

Problem Statement

Given an acute triangle ABCABC, mark 33 points X,Y,ZX, Y, Z in the interior of the triangle. Let X1,X2,X3X_1, X_2, X_3 be the projections of XX to BC,CA,ABBC, CA, AB respectively, and define the points Yi,ZiY_i, Z_i similarly for i=1,2,3i=1, 2, 3.
a) Suppose that XiYi<XiZiX_iY_i<X_iZ_i for all i=1,2,3i=1,2,3, prove that XY<XZXY<XZ.
b) Prove that this is not neccesarily true, if triangle ABCABC is allowed to be obtuse.
Proposed by Ivan Chan Kai Chin
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