# Fun with Numbers #2

Remember back to grade school geometry when your teachers forced you to memorize the Pythagorean theorem? wonder why it works?

Imagine triangle with the legs being lengths $a$ and $b$ and the hypotenuse (the longest side) $c$. Now we get four of those triangles for form a square.

We know the area of the outer square is $(a+b)^2$ but what about the interior square? What is the interior square compared to the outer one?

To find the area of the internal square we subtract the area of the four triangles. Each triangle has an area of $\frac{1}{2}ab$, as a result we have :

$(a+b)^2 - 4( \frac{1}{2}(ab)) = c^2$

$a^2 + 2ab + b^2 - 2ab = a^2 + b^2 = c^2$

Groups of three numbers such as $\{3,4,5\}$ and $\{5,12,13 \}$ are called Pythagorean triples because they satisfy the Pythagorean theorem. It is possible to generate Pythagorean triples using a set of equations called Euclid’s formula.