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Let O be the circumcenter of the triangle ABC

Source: Balkan MO 1999, Problem 1

April 24, 2006
geometrycircumcirclevectorgeometric transformationhomothetyratiotrapezoid

Problem Statement

Let OO be the circumcenter of the triangle ABCABC. The segment XYXY is the diameter of the circumcircle perpendicular to BCBC and it meets BCBC at MM. The point XX is closer to MM than YY and ZZ is the point on MYMY such that MZ=MXMZ = MX. The point WW is the midpoint of AZAZ. a) Show that WW lies on the circle through the midpoints of the sides of ABCABC; b) Show that MWMW is perpendicular to AYAY.
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