Question

Find a general formula for three integers which may be taken to represent the lengths of the sides of a right-angled triangle.

Answer 1

By pythagoras;-

$x^2+y^2=z^2$ or $x^2=x^2-y^2$

Put $mx=n(z+y)$ and $nx=m(z-y)$

Hence it is seen that;-

$\frac{x}{2mn}=\frac{y}{m^2-n^2}=\frac{z}{m^2+n^2}$

Hence the sides are;-

$2mn,m^2-n^2$ and hypotenuse $m^2+n^2$