MathDB

In triangle

ABC

, AB \equal{} 10, BC \equal{} 14, and CA \equal{} 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\minus{}b\sqrt{c}, where

a

b

, and

c

are positive integers and

c

is not divisible by the square of any prime. Find a\plus{}b\plus{}c.

Problem Statement