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P<= ( r+\sqrt{r^2+4R^2})/2 for bicentric quadr.

Source: VI All-Ukrainian Tournament of Young Mathematicians, Qualifying p14

May 25, 2021

geometrybicentric quadrilateralgeometric inequalityUkrainian TYM

Problem Statement

A quadrilateral whose perimeter is equal to

P

is inscribed in a circle of radius

R

and is circumscribed around a circle of radius

r

. Check whether the inequality

P\le \frac{r+\sqrt{r^2+4R^2}}{2}

holds. Try to find the corresponding inequalities for the

n

-gon (

n \ge 5

) inscribed in a circle of radius

R

and circumscribed around a circle of radius

r