Inspired by Joel Tan

Geometry Level 5

Let \(x, y, z\geq 0\) be reals such that \(x+y+z=1\) where points \(A(x+y,x^2+y^2), B(x+z,x^2+z^2)\), and \(C(y+z,y^2+z^2)\) are the vertices of a triangle.

If the maximum area of \(\Delta ABC\) can be expressed in simplest form as \(\frac{a}{b\sqrt{c}}\) for coprime positive integers with \(c\) square-free, what is the value of \(a+b+c\)?

Inspiration

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