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Perpendicular bisector

Source: Romania TST 2009, Day 3, Problem 3

July 26, 2009
geometrycircumcirclegeometric transformationreflectionEulerincentertrigonometry

Problem Statement

Let ABC ABC be a non-isosceles triangle, in which X,Y, X,Y, and Z Z are the tangency points of the incircle of center I I with sides BC,CA BC,CA and AB AB respectively. Denoting by O O the circumcircle of △ABC \triangle{ABC}, line OI OI meets BC BC at a point D. D. The perpendicular dropped from X X to YZ YZ intersects AD AD at E E. Prove that YZ YZ is the perpendicular bisector of [EX] [EX].
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