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circumradii R_A + R_C > R_B + R_D iff angles works out

Source: IMO Shortlist 1996, G8

August 9, 2008
inequalitiestrigonometrygeometrycircumcircleIMO Shortlist

Problem Statement

Let ABCD ABCD be a convex quadrilateral, and let RA,RB,RC,RD R_A, R_B, R_C, R_D denote the circumradii of the triangles DAB,ABC,BCD,CDA, DAB, ABC, BCD, CDA, respectively. Prove that R_A \plus{} R_C > R_B \plus{} R_D if and only if \angle A \plus{} \angle C > \angle B \plus{} \angle D.
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