generalized Pythagorean theorem

Theorem 1

If three similar polygons are constructed on the sidesof a right triangle, then the area of the polygon constructed on the hypotenuse isequal to the sum of the areas of the polygons constructed on the legs.

We when say that the polygon is constructed on a side of the right triangle, we meanthat the polygon shares an entire side with the polygon.

Beginning of proof.First, it suffices to prove the theorem for polygons of only one shape.Suppose that the areas of two polygons $P$ and $P^{\prime }$ of different shapesconstructed on some side of the triangle have a ratio $k$. Then the areasof polygons similar to them (say $R$ and $R^{\prime }$) and constructed on another sidewhich is $m$ times longer, will be $m^2$ times larger for both shapes. Therefore,they will have the same ratio, $k$. Hence if the areas of $P^{\prime },Q^{\prime },R^{\prime }$ satisfythe property that the first two add up to the third one, then the samewill hold true for the areas of $P,Q$ and $R$ where are $k$ times greater.

So instead of constructing a square on each side, as Euclid did, we use a righttriangle that is similar to the original right triangle. And instead of constructingthe triangle on the outside, we use the inside of the triangle.

Drop an altitude of the right triangle to its hypotenuse. This altitude divides thetriangle into two triangles and each is similar to the original triangle.We now have three similar right triangles constructed on the sides of the originalright triangle, and two of them add up to the third one.

End of proof.