Sheesh, how have I lived this long and not seen this proof of the Pythagorean Theorem?
Here’s a right triangle, with legs and and hypotenuse .Here’s the same triangle scaled up by a factor of , so the legs are and and the hypotenuse is .And here it is twice more, scaled up by factors of …… and .Of course these all are similar triangles.
Now, notice the hypotenuse of the first scaled-up triangle is the same length as one leg of the third, so we can draw them juxtaposed like this:Capital letters label the vertices.
Since the triangles are similar, angles and are congruent, and since they’re right triangles, angles and add to a right angle; therefore angle is a right angle.
Similarly the hypotenuse of the second scaled triangle is congruent to the other leg of the third, so they can be juxtaposed as well:and as before, angle is a right angle. Also, angles and sum to a right angle so angle is two right angles, i.e., , , and are colinear. Therefore is a rectangle, and . But and . So . QED.
What I like about this proof is how easy it is to grasp visually. Most proofs of the Pythagorean Theorem deal with areas, specifically persuading you the areas of two squares sum to the same as the area of a third square, but summing areas isn’t easy to visualize. Here there are no areas, just lengths, and we all can see immediately that is just .
(I saw this proof on, um, some blog recently. And failed to make note of exactly where. Sorry.)