## General Solution of Pythagoras Theorem

Pythagoras Theorem states that for a right angled triangle, with sides labelled as shown,
$a^2+b^2=c^2$

There is a general rue for satisfying all the sets of sides that satisfy Pythagoras Theorem.
It is
$a=2mn, \: b=m^2-n^2, c=m^2+n^2, \: m \gt n$
.
We can show that these satisfy Pythagoras Theorem.
\begin{aligned} a^2+b^2 &= (2mn)^2+(m^2-n^2)^2 \\ &= 4m^2n^2+m^4-2m^2n^2+n^4 \\ &= m^4+2m^2n^2+n^4 \\ &= (m^2+n^2)^2 \\ &= c^2 \end{aligned}