Let
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
be a convex quadrilateral, and let
![R_A, R_B, R_C, R_D](/media/m/a/6/e/a6e29b592ca2a6958ddcc00291019ab4.png)
denote the circumradii of the triangles
![DAB, ABC, BCD, CDA,](/media/m/8/a/e/8aefa92859fcd2d8e374ef5b700320d0.png)
respectively. Prove that
![R_A + R_C > R_B + R_D](/media/m/e/f/9/ef96979d15bd72470d3292c1af5f9bbf.png)
if and only if
%V0
Let $ABCD$ be a convex quadrilateral, and let $R_A, R_B, R_C, R_D$ denote the circumradii of the triangles $DAB, ABC, BCD, CDA,$ respectively. Prove that $R_A + R_C > R_B + R_D$ if and only if $\angle A + \angle C > \angle B + \angle D.$