# 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 $$A + B + C + D + E$$.

×