Da Vinci’s proof of the Pythagorean Theorem is essentially the following picture:
The details of what you’re looking at are as follows:
Take a right triangle (in black/gray) and augment each of its sides , , and by adding the squares , , and , respectively, of side lengths , , and , respectively. Let , , and . The goal, of course, is to show that .
To do this, we let and we construct two auxiliary quadrilaterals and (where here, and denote the – and -coordinates, respectively, of a point in the plane).
The claim is that and are equal, which we deduce by proving that is congruent to Geometrically, this follows from the fact that quadrilateral is nothing more than reflected about the segment and rotated 90° about .
Of course, this can also be argued rigorously using analytic/Euclidean-type arguments: See, e.g., here.
The end result is the same either way, of course: as claimed. Now, we can decompose the areas and :
In particular, yields the desired result: .
Some people may not be happy with the filling in of in the above blocked expressions: Visually, it’s not hard to believe that they decompose in such a way but Euclid would surely object. I myself am not Euclid, though I may go through and try to make some of these hand-wavey bits more Euclidean at some point.