Some Bonnesen-Style Triangle Inequalities

Abstract

Some Bonnesen-style isoperimetric inequalities for triangles in the plane are presented. For example, it is shown that $L^2-12 \sqrt 3 A \geq 35.098 \,\,\, r(R-2r)$ for triangles with perimeter $L$, area $A$, inradius $r$, and circumradius $R$. Equality holds when and only when either the triangle is equilateral or the triangle is similar to the isosceles triangle with sides 1, 1, and $\lambda$ where $\lambda\approx 1.23628634$ is the largest root of the equation $31x^3-28x^2-16x+4=0$.