How physicists prove the Pythagorean theorem

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How physicists prove the Pythagorean theorem.

If the hypotenuse and acute angle of one adjacent triangle are equal to the hypotenuse and acute angle of another adjacent triangle, then such triangles are congruent. This means that the hypotenuse and the acute angle completely define the triangle.

From the dimension messages it follows that the area of ​​the triangle $ABC$ (see figure) can be written as the square of the hypotenuse $c$ multiplied by some function of the angle $\alpha$: $c^2 f(\alpha)$. A similar operation can be carried out for similar adjacent triangles $ACD$ and $BCD$, for which the role of the hypotenuse is played by the category of the original triangle. Since the area of ​​triangle $ABC$ is equal to the sum of the areas of triangles $ACD$ and $BDC$, then

$$c^2 f(\alpha) = a^2 f(\alpha) + b^2 f(\alpha),$$

where $f(\alpha)$ is the same dimensionless function of the angle $\alpha$. From here

$$c^2 = a^2 + b^2.$$