Sum of squares

If a^2 + b^2 = x^2 + y^2 (and a, b, x, y all not zero)

then I am 99% sure either a = x, b = y or a = y, b = x. But what
makes this true, because it is clearly not true if we remove the
squares:

2 + 5 = 1 + 6

Geometrically, of course, a^2 + b^2 = r is the equation of a circle.
So if x^2 + y^2 = r as well, both equations represent the same circle.

But I would like a symbolic proof without appealing to geometry.

.

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