Question 15
In triangle $ABC,$ $AB = 10,$ $BC = 14,$ and $CA = 16.$ Let $D$ be a point in the interior of $\overline{BC}.$ Let $I_B$ 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 $a + b + c.$