The sides $$x$$ and $$y$$ of a scalene triangle satisfy $$x + \frac{2\Delta}{x} = y + \frac{2\Delta}{y}$$, where $$\Delta$$ is the area of the triangle. If $$x = 60$$, $$y = 63$$, what is the length of the largest side of the triangle?

Rearranging gives $$x - y = 2\Delta \cdot \frac{x-y}{xy}$$, and since the triangle is scalene $$x \neq y$$, so $$2\Delta = xy = 3780$$. But the area equals $$\frac{1}{2}xy \sin C$$ where $$C$$ is the angle between these two sides, so $$\sin C = 1$$ and $$C = 90^\circ$$. The largest side is the hypotenuse $$\sqrt{60^2 + 63^2} = \sqrt{7569} = 87$$.