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XZ passes through the midpoint of BK, isosceles, KX = CX, angle bisector

Source: 1st Girls in Mathematics Tournament 2019 p5 (Brazil) / Torneio Meninas na Matematica (TM^2 )

May 25, 2020
geometrymidpointisoscelesequal segmentsangle bisectorprojective geometry

Problem Statement

Let ABCABC be an isosceles triangle with AB=ACAB = AC. Let XX and KK points over ACAC and ABAB, respectively, such that KX=CXKX = CX. Bisector of ∠AKX\angle AKX intersects line BCBC at ZZ. Show that XZXZ passes through the midpoint of BKBK.
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