The Pythagorean Theorem - Picture Proof

Most people with a basic education in Mathematics will know and have used the Pythagorean Theorem (note properly referred to as this and not `Pythagoras Theorem` or `Pythagoras`s Theorem` since we are not certain it was Pythagoras himself who this theorem is due to). It states the following:

Given a right angled triangle, with sides a and b extending from the right angle and hypotoneuse c, then it follows:

`a` squared plus `b` squared equals `c` squared.

or if you prefer, the sum of the squares of the two shorter sides equals the square of the hypotoneuse.

What is more important in advanced Mathematics however is not the what but the why, why is this the case given ANY right angled triangle???

To answer this we need a proof, a logical argument and series of results linking our starting point to our desired conclusion.

To begin with, we consider the following:

A right angled triangle, with sides a,b,c as described previously. N.B. a,b,c > 0 |\ | \ a | \ c |_ _\ b

Create the following picture, with four copies of this triangle...

_b_ | _ _a_ | / \ | a| /c c\ |b |/ - - /| b| \ c c/ |a |_ _\_ |/_ _| a b

We see easily that the outer shape is a square with side length (a+b), furthermore it is not too challenging to show that the interior shape is also a square with side length `c`. All you really need to do is show that one of the interior angles must be 90 degrees, and the other three will follow from generality. To prove that it is 90 degrees, we combine that there are 180 degrees in a straight line with there being 180 degrees in a trangle. This step may be a good exercise for the reader.

Now to follow through to our desired conclusion, we form up equivalent expressions for the area of this shape.

On the one hand it is (a+b) x (a+b), simply the area of a square with side length (a+b).

Knowing the area of a trianlge, we may also formulate the area of this shape as being the sum of the areas of the four surrounding triangles and the interior square, this is...

4 x 0.5 x b x a + c x c.

Equating these expressions, we have and simplify the following equation:

(a+b)(a+b) = 2ab + cc

aa + 2ab + bb = 2ab + cc

aa + bb = cc

That is `a` squared plus `b` squared equals `c` squared, and thus we have proven the Pythagorean Theorem.

I hope anyone reading this who does not know this proof will take pleasure in its simplicity, that on a whim you may recall and demonstrate this proof to others.