Problem

Prove the the Pythagoras' Theorem, using Euclid's proof.

Solution

Pythagoras' Theorem states that in a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

Euclid's proof is as follows:

Let ABC be a right angled triangle, with the right angle at C.

Draw the square on each side, and label the length of each side as shown.

Now we have four congruent triangles:

We can see that:

AC = A'C'

BC = B'C'

AB = A'B'

Therefore:

AC^2 = (A'C')^2

BC^2 = (B'C')^2

AB^2 = (A'B')^2

Now we can see that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.