T-1 · Pythagoras

a² + b² = c², measured.

Drag the ends of the legs. The square on the hypotenuse is measured from the coordinates of its four corners, not from the formula, and set against a² + b². Then watch four triangles prove it, and follow the theorem off the page into a box.

Leg aLeg b

The theorem

(3, 4, 5) is a primitive Pythagorean triple: whole numbers, no common factor.

a² + b²	9 + 16 = 25
c	5
square on c	25 by the shoelace sum over its four corners
c² − (a² + b²)	residual …

Euclid's formula

For whole numbers m > n > 0, the legs m² − n² and 2mn have hypotenuse m² + n². The triple is primitive exactly when m and n share no factor and one of them is even.

m² − n²	4 − 1 = 3
2mn	2 · 2 · 1 = 4
m² + n²	4 + 1 = 5
check	9 + 16 = 25 residual …
primitive	yes gcd(m, n) = 1, one even