114 Chapter 10

Next, construct the perpendicular at the new lower right vertex, forming the green trian-

gle. This line meets the opposite side, because of our assumption that the rectangle was

not more than twice as tall as it is wide (why?). By translating the green triangle, we form

a rectangle. Since one side is s and the area is the same as the original rectangle, the other

side must also be s, and so this is the desired square.

10.5 Combining squares

We shall now seek to combine the squares together.

Lemma 77.4. Any two squares joined together are dissection congruent with a single

larger square.

Proof. This argument amounts to one of the classical dissection proofs of the Pythagorean

theorem.

→ →

Namely, place the two squares adjacent to each other as shown, and slice off two right

triangles whose legs are congruent with the sides of the squares. The triangles can be

reassembled as shown to form a larger square.

In the arguments above, we were actually making a subtle use of the following fact,

which I should like now to highlight.

Observation 78. The dissection congruence relation is transitive: if P is dissection con-

gruent to Q and Q is dissection congruent to R, then P is dissection congruent to R.

Proof. We may cut P into finitely many pieces and rearrange them in order to form poly-

gon Q, and we may cut Q into pieces so as to form R. The cuts we make in Q induce

cuts on the individual pieces that we had already made when forming Q out of P. Itisas

though the individual pieces we had made in P are cut again to form smaller pieces. These

subpieces can be seen to provide a partition of P that can be used to form R.