Pythagorean Theorem: My Favourite Proof

by Hellotherein Mathematicson Posted on

The Pythagorean Theorem is of course a well-known theorem. Basically, it states that the sum of squares of the lengths of the two legs of a right triangle is equal to the square of the length of the hypotenuse. There are many proofs of this. The following is perhaps my favourite.

Consider the diagram below:

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The area of the large square is (a+b)^2. The area of each yellow triangle is ab/2. The area of the green square is c^2. The large square consists of 4 yellow triangles and the middle green square. Thus, we have:

    \begin{align*} (a+b)^2 &= 4\big(\frac{ab}{2}\big)+c^2 \\ \Rightarrow a^2+2ab+b^2 &= 2ab + c^2 \\ \Rightarrow a^2 + b^2 &= c^2. \end{align*}