Dissection

The Sliding Squares

Traditional · Antiquity · DissectionDifficultyPen & paper

The square on each leg can be slid — sheared, not cut — until it exactly fills its matching rectangle inside the square on the hypotenuse, with no change in area at any point along the way.

Start with the two leg-squares sitting in their natural place, touching corner to corner at the right angle $A$ .
Drop a perpendicular from $A$ to the hypotenuse $B C$ , and carry it on to $L$ on the far side of the square on $B C$ . This splits that square into two rectangles.
Take the square on $A B$ and lean it over — a shear, sliding its top edge sideways while its base stays put — until its far edge lines up with the near edge of the first rectangle.
A shear like this never changes area: the base and the perpendicular height between the two parallel sides stay fixed throughout the slide, no matter how far it leans. So the leaned square still has area $A B^{2}$ — and it now exactly covers the rectangle.
Shear the square on $A C$ the same way into the second rectangle. Both leg-squares have now poured, without gaining or losing area, into the two pieces of the square on $B C$ .

Therefore c² = a² + b². ∎

Shearing is not cutting: nothing here is severed and rejoined. It’s the same argument Euclid makes with congruent triangles (see The Bride’s Chair), told instead through a continuous slide — a reminder that “same base, same parallels, same area” is really one idea wearing two costumes.