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configuration in triangle with radii

Source: Serbia MO 2005 2nd Grade P3

April 11, 2021
geometryTriangle

Problem Statement

In a triangle ABCABC, DD is the orthogonal projection of the incenter II onto BCBC. Line DIDI meets the incircle again at EE. Line AEAE intersects side BCBC at point FF. Suppose that the segment IO is parallel to BCBC, where OO is the circumcenter of △ABC\triangle ABC. If RR is the circumradius and rr the inradius of the triangle, prove that EF=2(R−2r)EF=2(R-2r).
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