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Can you prove the Pythagorean Theorem?


The famous Pythagorean Theorem deals with problem of diagonal measure. It was known to the Egyptians and the Chinese (circa 1500 BCE), over one-thousand years before the Greeks, but the Pythagoreans were able to prove it to be true. Despite the fact that it is sometimes called Pythagoras' Theorem, it is not certain if Pythagoras actually discovered it. Pythagoras of Samos (569-500 BCE) established a cult called the Pythagorean brotherhood and its members were sworn to absolute secrecy and found themselves under threat of death if they divulged any of the cult's secrets. It was not until many years after Pythagoras' death that the famous theorem came to light. For this reason we cannot be certain if he or one of his disciples discovered and proved the result. Although I shall use different methods to the Pythagoreans (documented in Euclid's Elements), I shall demonstrate the result by two different, but related, methods.

Theorem
The square on the long side of any right triangle is equal to the square on the other two sides.

    a² + b² = c².

Proof 1
Consider the diagram below.

Clearly the area of the sloping square is c². But we can approach it by a different means.

Area of surrounding square = ( a + b )² = a² + b² + 2ab
Area of rectangle = ab
Area of four triangles = Area of two rectangles = 2ab
Area of sloping square = a² + b² + 2ab – 2ab = a² + b².

Hence c² = a² + b², or the square on the long side of the right triangle is equal to the squares on the other two sides.

Proof 2
This result can be demonstrated much more elegantly using the following set of diagrams. Watch the right triangle in the bottom right corner carefully.

The area of the shaded region, which is currently a square, must be c². By sliding the triangle as indicated, we obtain the following diagram.

Obviously by moving the triangle the shaded area remains unchanged and will still be c². So we continue by sliding the other two triangles as shown.

From this final diagram we can see that the shaded region is equal to a² + b², in other words c² = a² + b².