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Cyclic quadrilateral ABCD with CD^2=AD.ED; E=DA∩CB

Source: Czech-Polish-Slovak Match 2007-P3

September 14, 2011
geometrycircumcirclegeometry proposed

Problem Statement

A convex quadrilateral ABCDABCD inscribed in a circle kk has the property that the rays DADA and CBCB meet at a point EE for which CD^2=AD\cdot ED. The perpendicular to EDED at AA intersects kk again at point F.F. Prove that the segments ADAD and CFCF are congruent if and only if the circumcenter of ABE\triangle ABE lies on ED.ED.