OK, I can’t remember the name, but as I recall, the Pythagorean theorem can be considered a specific case of a more general theorem. That is, if you draw some shape along one side of a right triangle, then make that same shape with the same proportions along the other two sides, then the two smaller areas add to the side of the largest one. What is that theorem/property/principle called?
Or maybe I’m just misunremembering sumthin’.
The law of cosines.
Cool! Thanks!
While the Law of Cosines is, indeed, a generalization of the Pythagorean Theorem, I suspect that the OP is thinking of this:
*The Pythagorean theorem was generalized by Euclid in his Elements:
If one erects similar figures (see Euclidean geometry) on the sides of a right triangle, then the sum of the areas of the two smaller ones equals the area of the larger one.*
Some nifty-looking demonstrations and discussion of this fact can be found on this page.
Yep, that’s it.
A pretty trivial generalization, once you have the plain Pythagorean theorem and realize that similar figures whose dimensions are in the ratio a:b have areas in the ratio a^2:b^2 [i.e., blowing up dimensions by a factor of K means blowing up area by a factor of K^2].
There’s also Fermat’s Last Theorem.
While related in a way to Pythagorean triples, it could hardly be called a generalization of the Pythagorean theorem.