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geometric inequality

Source: Taiwan 3rd TST, 1st independent study, problem 1

August 12, 2005
inequalitiesgeometrycircumcirclegeometry proposed

Problem Statement

Let PP be a point in the interior of △ABC\triangle ABC. The lengths of the sides of △ABC\triangle ABC is a,b,ca,b,c, and the distance from PP to the sides of △ABC\triangle ABC is p,q,rp,q,r. Show that the circumradius RR of △ABC\triangle ABC satisfies R≤a2+b2+c218pqr3.\displaystyle R\le \frac{a^2+b^2+c^2}{18\sqrt[3]{pqr}}. When does equality hold?
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