This equation emerged while I was solving a geometry problem, which was equivalent to proving $x+n=m$.
Solve the system of polynominal equations where $x,y,z,m,n,l$ are positive reals.
$$ \begin{cases} yz=nl\\ x^2=m(m+n)\\ (x+z)^2=m(m+n+l)\\ (x+z)^2+z^2=(m+n)^2\\ (x+z)^2+(y+z)^2=(x+y)^2 \end{cases} $$
Through complex calculation, I solved the equation for $z$, which gave me $x=\frac{\sqrt {7}+1}{3}z$, $m=\frac{\sqrt {7}+1}{2}z$, $n=\frac{\sqrt {7}+1}{6}z$.
However, this way seemed a bit complex, and solving it for other variables did not seem simpler either.
Is there any simple way to prove $x+n=m$ without calculating each value?