Point in an Isosceles Triangle

Triangle ABCABC is isosceles with a right angle at AA. A point is chosen uniformly at random inside the triangle. Let pp be the probability that the point is closer to the point AA than it is to BB or CC, and it is closer to side BCBC than it is to side ABAB or ACAC. pp can be expressed as abc\sqrt{\frac{a}{b}} - c, where aa and bb are coprime positive integers and cc is a positive integer. What is the value of a+b+ca + b + c?

Details and assumptions

It follows from the conditions that AB=AC AB = AC .

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