MATHEMATICS AND ORIGAMI

Mathematics and Origami

3 Any scalene right triangle ABC

From ∆abc we ́ll construct the exterior square with side CK = a+b. Then we fold that

square to get the interior square ABIF. If the Pythagorean theorem is fulfilled for ∆ABC, it

follows that the area of square ABIF will be equal to the sum of AKJE and EBLM.

The square of side c is made up by 4 triangles like ABC plus the central square with

side b + a – 2a = b - a.

At the same time, the square of side b is made up by said central square, two ∆ABC and

a rectangle with sides a, b – a.

Therefore, the difference between squares with sides c and b is:

4 ∆ABC - 2∆ABC – a (b – a)

()^22

2

2 ab a ab ab a a

ab

− − = − + =

i.e. the difference between the square built on the hypotenuse and that constructed on the great
leg, is the square on the small leg: this is also a way of enunciating the Pythagorean theorem.

C

K A

J

E

B

L

M I

F

a

b

c

Interlude