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|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|, circumradii, inradii

Source: Estonia IMO TST 2007 p2

March 28, 2020
circumradiusinradiusaltitudegeometryGeometric Inequalitiescircumcircleincircle

Problem Statement

Let DD be the foot of the altitude of triangle ABCABC drawn from vertex AA. Let EE and FF be points symmetric to DD w.r.t. lines ABAB and ACAC, respectively. Let R1R_1 and R2R_2 be the circumradii of triangles BDEBDE and CDFCDF, respectively, and let r1r_1 and r2r_2 be the inradii of the same triangles. Prove that ∣SABD−SACD∣>∣R1r1−R2r2∣|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|
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