Pythagoras is not a fact about triangles
Hold the legs at 3 and 4, hold the angle between them at exactly 90°, and draw the triangle on a sphere instead of a page. The hypotenuse comes out at 4.57.
Take a right triangle with legs of 3 and 4. The hypotenuse is 5, and it has been 5 for as long as anyone has been asking. Now keep both legs at 3 and 4, keep the angle between them at exactly 90°, and draw the same triangle on the surface of a sphere. The hypotenuse is no longer 5. On a sphere of radius 3.62 it comes out at about 4.57. Draw it on a hyperbolic surface of the same radius and it comes out at about 5.31.
Nothing about the triangle changed. The right angle is still a right angle; the two legs are still 3 and 4. What changed was the space.
What replaces it
On a sphere of radius R the relation between the three sides of a right triangle is
cos(c/R) = cos(a/R) · cos(b/R)
and on a hyperbolic surface it is the same sentence with the hyperbolic functions:
cosh(c/R) = cosh(a/R) · cosh(b/R)
Neither of these looks like a² + b² = c². But expand either one as a series in small a/R and b/R — that is, let the surface get large compared with the triangle — and the first term that survives is the square. Both collapse to a² + b² = c². The familiar theorem is the limiting case of both curved laws at once, which is a strange and rather beautiful position for something to occupy.
This is not a curiosity about exotic geometry. It is the reason the theorem is equivalent to Euclid's parallel postulate: assume either one and you can prove the other. For two thousand years mathematicians tried to derive the parallel postulate from Euclid's other four axioms, and the reason they failed is that it is not derivable. It is a choice, and choosing it is the same act as asserting that a² + b² = c².
The second symptom
The hypotenuse is not the only thing that moves. Watch the angles.
On the sphere, that 3-4 triangle has angles of about 50.7° and 69.7° alongside its right angle, a total of 210.3°, when a flat triangle would give 180°. On the hyperbolic surface the same two legs give 26.8° and 40.9°, totalling 157.7°. The angle sum overshoots on a sphere and falls short in hyperbolic space, and it hits 180° in exactly one place.
The two symptoms are not independent. On a surface of constant curvature, with the three sides taken along geodesics, the amount by which the angle sum misses 180° is exactly the curvature multiplied by the area of the triangle. That is the Gauss–Bonnet theorem in its simplest form; in general it integrates the curvature over the region rather than multiplying by it. So on a surface whose curvature you already know, the three angles of a triangle fix its area. No lengths required.
The Pythagoras Lab does exactly this. Drag the curvature slider and both readings move together while the legs and the right angle stay pinned.
Why anyone should care
A surveyor measuring a building site can use a² + b² = c² without a second thought; over a hundred metres the curvature of the Earth contributes an error far below the precision of the tape. A geodesist laying out a national survey network cannot. The error in the flat answer grows with the cube of the size of the triangle: double every side and it multiplies by eight. That is why triangulation networks covering a country have always been computed with spherical or ellipsoidal formulas, and why the mathematics in a GPS receiver is not the mathematics in a school textbook.
And in general relativity the curvature is not a property of a surface embedded in space; it is a property of spacetime itself, sourced by mass and energy. The statement "space is flat here" is a physical claim about the local distribution of matter, testable by measuring triangles. That is the sense in which Pythagoras is an empirical proposition rather than a mathematical certainty: it holds where spacetime is flat, and spacetime is not flat everywhere.
The proof is about shape, not size
Given how much the theorem depends on flatness, the flat proof of it is remarkably cheap.
Drop a perpendicular from the right angle onto the hypotenuse. It lands somewhere, splitting the hypotenuse of length c into two pieces, p and q. Look at the two smaller triangles this creates: each has a right angle, and each shares one of its other angles with the original triangle. Three angles that agree means the triangles have the same shape: they are similar to the original and to each other.
Similar triangles have proportional sides. Apply that to each piece and you get a² = c·p and b² = c·q. Add them: a² + b² = c(p + q) = c². The entire theorem is one perpendicular line plus the fact that equal angles force proportional sides.
That last fact is itself Euclidean, which is the loop closing. On a sphere there are no similar triangles of different sizes at all: fix the three angles and you have fixed the area, so you cannot scale a triangle without changing its shape. The proof and the theorem it proves stand or fall together, and both stand only on a flat surface.
You can watch each step in the Similar Triangles Lab, which is where that Euclidean fact gets tested: drag the scale factor from a quarter to six, and the three angle readings refuse to move.
A footnote about squares
One more thing the theorem does not require: squares.
Build any shape you like on the three sides — semicircles, pentagons, silhouettes of cats — and provided all three are similar to one another, the two on the legs still add up to the one on the hypotenuse. Euclid states this in Elements VI.31, and it holds for the same reason everything else here does: area scales as the square of length, so the ratio of the three figures is fixed by the ratio of the three sides no matter what the figures are.
Use semicircles and something startling falls out. The semicircle on the hypotenuse passes exactly through the right angle, and the two crescents left outside it have a combined area exactly equal to the triangle. Hippocrates of Chios found that around 440 BCE, and it was the first time anyone showed that a region with curved edges could have precisely the area of a straight-sided one — a genuine result, two millennia before calculus made such things routine.