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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