nonlinear diophantine equation $x^2+y^2=z^2$ how to solve a diophantine equation $x^2+y^2=z^2$ for integers $x,y,z$
i strongly believe there is a geometric solution ,since this is a pythagoras theorem form
or a circle with radius $z$
$x^2+y^2=z^2$
$(\frac{x}{z})^2+(\frac{y}{z})^2=1\implies x=y=\pm z$ or $0$
so we consider a line passing through  points $P_1(- z,0)$ and  $P(x,y)$ both on the circle
$m=\frac{y}{x+z}$
$x^2+m^2(x+z)^2=z^2$
$(m^2+1)x^2+2xzm^2+(m^2-1)z^2=0$
$((m^2+1)x+(m^2-1)z)(x+z)=0$
$\frac{x}{z}=-\frac{m^2-1}{m^2+1}$ or $-1$
let $m=\frac{a}{b}\implies \frac{x}{z}=\frac{b^2-a^2}{b^2+a^2}$
$\frac{y}{z}=\frac{2a^2}{b^2+a^2}$
how to get explicit $z,x,y$

Euclid's Formula says that in essence, $(m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2$ for all positive integers $m > n$. 
This is basically a parametrization of Pythagorean Triplets with two parameters.
