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prove that p(1 -2Rr) <=1, when sides a,b,c <=1

Source: 2015 Sharygin Geometry Olympiad Correspondence Round P15

August 2, 2018

geometryGeometric Inequalitiesinradiuscircumradius

Problem Statement

The sidelengths of a triangle

ABC

are not greater than

1

. Prove that

p(1 -2Rr)

is not greater than

1

, where

p

is the semiperimeter,

R

and

r

are the circumradius and the inradius of

ABC