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Nice geometry from MMO

Source: Morrocan TST 2005 (pb 4)

May 27, 2005
geometrycyclic quadrilateralperpendicular bisectorgeometry proposed

Problem Statement

Consider a cyclic quadrilateral ABCDABCD, and let SS be the intersection of ACAC and BDBD. Let EE and FF the orthogonal projections of SS on ABAB and CDCD respectively. Prove that the perpendicular bisector of segment EFEF meets the segments ADAD and BCBC at their midpoints.
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