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Ttriangles with the same area

Source: Turkey TST 2014 Day 1 Problem 3

March 12, 2014
geometrycircumcircleparallelogramgeometry proposed

Problem Statement

Let r,Rr,R and rar_a be the radii of the incircle, circumcircle and A-excircle of the triangle ABCABC with AC>ABAC>AB, respectively. I,OI,O and JAJ_A are the centers of these circles, respectively. Let incircle touches the BCBC at DD, for a point E∈(BD)E \in (BD) the condition A(IEJA)=2A(IEO)A(IEJ_A)=2A(IEO) holds. Prove that ED=AC−AB  ⟺  R=2r+ra.ED=AC-AB \iff R=2r+r_a.
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