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The kinetic energy formula $E=\frac{1}{2}mv^2$ relates energy to the square of the velocity.

There are many nice explanations on this site which deal with this fact.

But could we give an explanation by the fact that on the Euclidean space the Pythagorean theorem holds?

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The kinetic energy formula is not related to the Pythagorean theorem, so the answer to the title question is "no".

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