# Point in an Isosceles Triangle

Triangle $$ABC$$ is isosceles with a right angle at $$A$$. A point is chosen uniformly at random inside the triangle. Let $$p$$ be the probability that the point is closer to the point $$A$$ than it is to $$B$$ or $$C$$, and it is closer to side $$BC$$ than it is to side $$AB$$ or $$AC$$. $$p$$ can be expressed as $$\sqrt{\frac{a}{b}} - c$$, where $$a$$ and $$b$$ are coprime positive integers and $$c$$ is a positive integer. What is the value of $$a + b + c$$?

Details and assumptions

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

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