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Prove that two lines are perpendicular

Source: Turkish TST 2011 Problem 2

July 23, 2011
geometryincentercircumcirclegeometric transformationreflectionrhombussymmetry

Problem Statement

Let II be the incenter and ADAD be a diameter of the circumcircle of a triangle ABC.ABC. If the point EE on the ray BABA and the point FF on the ray CACA satisfy the condition BE=CF=AB+BC+CA2BE=CF=\frac{AB+BC+CA}{2} show that the lines EFEF and DIDI are perpendicular.
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