generalized Pythagorean theorem

Theorem 1

If three similar polygons are constructed on the sides of a right triangle, then the area of the polygon constructed on the hypotenuse is equal 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 mean that 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 shapes constructed on some side of the triangle have a ratio $k$. Then the areas of polygons similar to them (say $R$ and $R^{\prime }$) and constructed on another side which 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 $P^{\prime },Q^{\prime },R^{\prime }$ satisfy the property that the first two add up to the third one, then the same will 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 right triangle that is similar to the original right triangle. And instead of constructing the triangle on the outside, we use the inside of the triangle.

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

End of proof.

Title	generalized Pythagorean theorem
Canonical name	GeneralizedPythagoreanTheorem
Date of creation	2013-03-22 17:13:58
Last modified on	2013-03-22 17:13:58
Owner	yogis (15158)
Last modified by	yogis (15158)
Numerical id	12
Author	yogis (15158)
Entry type	Theorem
Classification	msc 51-00
Synonym	Pythagorean theorem
Related topic	RightTriangle
Related topic	Polygon
Related topic	PythagorasTheorem
Related topic	ProofOfPythagoreanTheorem2