Sum of Squares as Product of Factors with Square Roots

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Theorem

$x^2 + y^2 = \paren {x + \sqrt {2 x y} + y} \paren {x - \sqrt {2 x y} + y}$

Proof

\(\ds \paren {x + \sqrt {2 x y} + y} \paren {x - \sqrt {2 x y} + y}\)

\(=\)

\(\ds x^2 - x \sqrt {2 x y} + x y + x \sqrt {2 x y} - \sqrt {2 x y} \sqrt {2 x y} + y \sqrt {2 x y} + x y - y \sqrt {2 x y} + y^2\)

\(\ds \)

\(=\)

\(\ds x^2 + \paren {x - x} \sqrt {2 x y} + 2 x y - 2 x y + \paren {y - y} \sqrt {2 x y} + y^2\)

\(\ds \)

\(=\)

\(\ds x^2 + y^2\)

$\blacksquare$