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inequality with circumradii of 4 triangles of convex ABCD

Source: 2009 Sharygin Geometry Olympiad Final Round problem 2 grade 9

July 26, 2018
inequalitiesconvex quadrilateralcircumradius

Problem Statement

Given a convex quadrilateral ABCDABCD. Let Ra,Rb,RcR_a, R_b, R_c and RdR_d be the circumradii of triangles DAB,ABC,BCD,CDADAB, ABC, BCD, CDA. Prove that inequality Ra<Rb<Rc<RdR_a < R_b < R_c < R_d is equivalent to 180o−∠CDB<∠CAB<∠CDB180^o - \angle CDB < \angle CAB < \angle CDB .
(O.Musin)
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