One of the most fundamental truths of Euclidean geometry – and, indeed, of the geometry of the real world, for all that its precision here is limited by the scale of the triangle in relation to the local curvature of space-time – describes a relationship among the sides of a right-angled triangle. It has been known since antiquity – it was known in ancient Indian cultures and (probably subsequently) in Mesopotamian cultures. More recently yet, the ancient Greek school of thought known as the Pythagoreans (notable for also sharing, with some Indian cultures, both vegetarianism and a belief in re-incarnation) attributed it to their (possibly mythical) founder, Pythagoras. It asserts that
The sum of the areas of the squares on the two orthogonal sides of a right-angle triangle is the area of the square on the third side.
The third side, facing the right angle, is called
the hypotenuse
of the right-angle triangle. (The addition
of an extra term in the cosine of the angle between two
sides generalises this rule to the case of
non-right angles.)
Note that, although the theorem is usually stated in terms of squares on the sides of the triangle, equivalent results inevitably follow for other similar figures on each side of the triangle, with the side taking the same rôle in each. For example, the areas of semicircles on the three sides (each as diameter) are simply π/2 times the areas of the squares; likewise, the regular n-gon for any integer n > 2, on each side and so on. As long as the area of the figure is proportional to that of the square on the side it is attached to, the corresponding result follows naturally from the usual one.
Given a right-angle triangle, I need to show you how you can satisfy yourself that it has the claimed property. I'll explain what to do with the right-angle triangle you've got, illustrating each step with a right-angle triangle of my own. Hopefully, that'll help you see how to apply what I say to yours; and, thereby, lead you to see that the same is true of yours.
So take your right-angle triangle and rotate it through a quarter turn about its right-angle corner (in either sense). You now have two copies of it, with each side of either perpendicular to the corresponding side of the other; and the perpendicular sides all meet in the corner about which you rotated. In particular, for each of the perpendicular sides of your triangle, this gives you two perpendicular copies of that side: complete these to form a square, in each case.
I'm now going to show that, if we translate the combination of those two squares through the displacements along the two copies of our hypotenuse, its images fit around the original – abutting it, with no overlap and no gaps. From this it follows that, using these two translations, this figure tessellates the whole plane. If that's immediately apparent to you, you can skip all the diagrams and reasoning below. Since the two translations used are the same two that tessellate the plane with a square on either hypotenuse, this figure's area must in fact be equal to the area of that square, which is exactly the result to be proven.
To show that the tessellation works, let's first translate our pair of squares through one hypotenuse repeatedly (using alternating colours for each copy, to help tell them apart):
I trust it's clear that you can keep doing this indefinitely, in each direction, with each copy's concave corner at one end of the chosen hypotenuse surrounding the next copy's corner at the other end of that hypotenuse. Now let's take that train of pieces and translate it parallel to the other hypotenuse:
As you can see, and as I'm sure you'll find if you carefully perform the same operations with any right-angle triangle, the copies of the figure never overlap or leave gaps; and we can clearly continue repeating the pattern in each direction indefinitely, thereby tessellating the plane.
Now, indeed, I'm appealing to an intuition that figures tessellated by the same pair of translations must have equal area; but we can equally use our tessellated pair of squares and cut out the square on one hypotenuse, to look at the fragments of each square that it gives us. The tessellation property inescapably implies that these fragments, translated through various combinations of the two hypotenuse-displacements, can be reassembled into the two squares on the perpendicular sides – as can readily enough be verified. (Of course, this in turn relies on a different intuition, that cutting a figure up and rearranging its pieces doesn't change area. The two intuitions are, of course, equivalent.)
There are many other proofs, but this is my favourite – I dreamed it up for myself (albeit I can't rule out having been prompted thereto by some long-forgotten encounter with it, or something similar; and it is equivalent to one of the standard rearrangement proofs) – which gives me a natural bias in its favour – and, for many years, hadn't seen it from anyone else until it got a quick cameo in Mathologer's video about an odd prime being a sum of two squares precisely if it is one more than a multiple of four. I do, whether or not due to that natural bias, consider it to be more elegant and to touch more closely the reasons for the truth of the result it proves.
The same Mathologer video also tells me (a.a +b.b).(c.c +d.d) = power(2, a.c +b.d) +power(2, a.d −b.c), apparently known to ancient Greeks; and it sounds interesting.
Finally – in case you thought I played a trick by my choice of which right-angle triangle to use, or suspect that the variants you've tried only worked because you didn't pick the quirky special case that'd break the reasoning – consider this animation (or a line-only variant), that varies the ratio of the perpendicular sides of the triangle through the full range of relative values possible:
Pythagoras's theorem enables us to define an addition on squares, pairwise, by using a side of each as a the perpendicular sides of a right-angled triangle, with the square on the hypotenuse serving as sum of the two squares. The construction is inherently symmetric, so the addition is commutative (a.k.a. Abelian). Because two sides and an angle of a triangle determine its remaining side and angles, the addition is cancellable. Suppose we add three squares in this way: does it matter what order we do the additions ? (i.e., is the addition associative ?)
