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IW_1+ IW_2 + IW_3>= 2R + \sqrt{2Rr} , incenter, circumcircle

Source: XI All-Ukrainian Tournament of Young Mathematicians, Qualifying p15

May 27, 2021

geometryincenterinequalitiescircumcircleUkrainian TYM

Problem Statement

Let

I

be the point of intersection of the angle bisectors of the

\vartriangle ABC

W_1,W_2,W_3

be point of intersection of lines

AI, BI, CI

with the circle circumscribed around the triangle,

r

and

R

be the radii of inscribed and circumscribed circles respectively. Prove the inequality

IW_1+ IW_2 + IW_3\ge 2R + \sqrt{2Rr.}