Just The Third Side

Geometry Level 5

Let \( \Delta \: ABC\) be a triangle with distinct integer sides, \(AB \: = \: 9\), and \(AC\: = \: 7\).

There exists a point \(P\) inside \( \Delta \: ABC\) such that the sides of \(\Delta \: BPC\) are also distinct integers.

Let \(x \) be the minimum area of \(\Delta \: BPC\) when the perimeter of \(\Delta \: ABC \) is at its minimum, and let \(y\) be the maximum area of \( \Delta \: BPC\) when the perimeter of \( \Delta \: ABC\) is at its maximum.

\( x +:y \) can be expressed in the form \( \frac{A}{B} (\sqrt{C} + D\sqrt{E}) \) where \(A\) and \(B\) are coprime positive integers, and \(C\) and \(E\) are square-free.

Find \( A + B + C + D + E\).

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