Pythagorean theorem

The Pythagorean theorem (Pythagoras’ theorem; in Chinese also the Gougu theorem) says that in a triangle with a right angle at $C$, the legs $a$ and $b$ and the hypotenuse $c$ satisfy $a^2 + b^2 = c^2$: the squares on the legs together have the area of the square on the hypotenuse. The converse holds too — a triangle with $a^2 + b^2 = c^2$ has a right angle at $C$. The page draws Euclid’s figure (Elements I.47) and lets $C$ leave the right angle, which turns the theorem into the law of cosines.

Each altitude, carried on through the square on the opposite side, cuts that square into two rectangles. The rectangle next to a vertex has area (the two sides meeting there) × the cosine of the angle there, and the two rectangles next to the same vertex — one in each square that meets there — are equal: hatched next to $A$, dotted next to $B$, yellow next to $C$. Adding the three squares’ pieces gives the law of cosines. At a right angle at $C$ the yellow pieces vanish and the altitude from $C$ cuts the square on $c$ into exactly $a^2$ and $b^2$. Off the right angle the yellow pieces are the difference, each $ab\cos C$: positive when $C$ is acute, negative when it is obtuse. Any piece is negative, and drawn outside its square, exactly when the angle at its vertex is obtuse; the other piece of that square is then larger than the square.

The three proofs answer three questions. Shear (Euclid) moves the piece of $a^2$ next to $B$ into the square on $c$: a shear keeps base and height, a quarter turn keeps everything, so every frame keeps the area; off 90° only that piece moves and the yellow ones stay behind. Dissection (Zhao Shuang’s figure, 3rd century) fills $c^2$ with four copies of the triangle and a square of side $|b - a|$, $4\cdot\tfrac12 ab + (b-a)^2 = a^2 + b^2$, and turns two copies about corners to fill $a^2$ and $b^2$. Similar splits the triangle along its altitude into two copies of itself on $a$ and $b$; similar figures’ areas scale with the square of a side, so semicircles, equilateral triangles or the triangle itself on the three sides add up just as the squares do.