Dissection

The Twin Hexagons

Leonardo da Vinci · c. 1500 · DissectionDifficultyPen & paper

Two hexagons — one wrapped around the two smaller squares, one wrapped around the larger — turn out to be exactly the same shape, and that alone forces $a^{2} + b^{2} = c^{2}$ .

Let triangle $A B C$ have its right angle at $C$ . Erect the usual three squares outward — on $C A$ , on $C B$ , and on the hypotenuse $A B$ .
Between the two leg-squares, right at $C$ , a $90°$ wedge sits empty: the two squares' own corners already account for $180°$ there, and the triangle's own right angle accounts for another $90°$ , leaving exactly one triangle-shaped gap. Rotate a second copy of the triangle into it, so $C G$ matches $C A$ and $C H$ matches $C B$ — the fit is exact.
The two leg-squares, the original triangle, and its rotated twin now trace a single six-sided figure. Its area is just the sum of its pieces: $a^{2} + b^{2} + ab$ .
Do the same thing again on the far edge of the square on the hypotenuse: attach a second copy of the triangle there, matching leg to matching side exactly as before.
Da Vinci's insight: this second figure — one square plus the doubled triangle — traces a hexagon congruent to the first. Congruent figures have equal area, so $a^{2} + b^{2} + ab = c^{2} + ab$ . The shared $ab$ cancels, leaving $a^{2} + b^{2} = c^{2}$ .

Therefore c² = a² + b². ∎

This proof is popularly credited to Leonardo da Vinci, and it earns the attribution: it never once appeals to algebra or to a ratio, only to the idea that two shapes built from identical parts, arranged into the same outline, must weigh the same in area. It shares a grandparent with The Bride’s Chair — both ultimately trust a pair of triangles to match exactly — but where Euclid slides a rectangle into place, Da Vinci spins a whole triangle into the gap left by two abutting squares.