2009 AIME I Problem 15

In triangle ABC, AB = 10, BC = 14, and CA = 16. Let D be a point in the interior of \overline{BC}. Let points I_B and I_C denote the incenters of triangles ABD and ACD, respectively. The circumcircles of triangles BI_BD and CI_CD meet at distinct points P and D. The maximum possible area of \triangle BPC can be expressed in the form a - b\sqrt {c}, where a, b, and c are positive integers and c is not divisible by the square of any prime. Find a + b + c.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.