The Lewis Carroll problem

This is a puzzle.

Lewis Carroll, the author of Alice in Wonderland, was also a mathematician. In his diary for 1890, he wrote the following statement:

2(x^2 + y^2) is always the sum of two squares where x and y are a pair of non-zero integers.

Usually, when people want to see if Lewis Carroll was correct, they use algebra.

For this puzzle, I am going to ask you for a geometric argument to show whether Carroll’s result is correct.

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