## Solution of problem 11306 in AMM

Let $a,b,$ and $c$ be the lengths of the sides of a nondegenerate triangle, let $p=(a+b+c)/2$, and let $r$ and $R$ be the inradius and circumradius of the triangle, respectively. Show that $\dfrac{a}{2}\cdot \dfrac{4r-R}{R}\leq\sqrt{(p-b)(p-c)}\leq \dfrac{a}{2},$

and determine the cases of equality.

My solution.