# 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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