Solve in $\mathbb Z^3$. 
Let $x$, $y$ and $z$ be integer numbers. Solve the following equation.
  $$x^2+y^2+z^2=45(xy+xz+yz)$$

My trying.
It's a quadratic equation of $z$ and we need $\Delta=n^2$ for an integer $n$,
but it gives a very ugly expression. 
Thank you!

For  Will Jagy, I am sorry!
Let $x-y=a$, $y-z=b$ and $z-x=c$.
Hence, $a^2+b^2+c^2\vdots11$ and $a+b+c=0$.
Thus, $a^2+ab+b^2\vdots11$, which says that $a\vdots11$ and $b\vdots11$ and $x\equiv y\equiv z(\mod11)$, 
which gives $x\vdots11$, $y\vdots11$ and $z\vdots11$ (if $x\equiv y\equiv z\equiv r(\mod11)$ then $r^2\vdots11$).
Id est,  an infinite descent ends this problem.
Done!
