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.

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