Stuck with an equation having 6 unknowns

Stuck with an equation having 6 unknowns

Suppose $a,b,c,d,e,f\in\mathbb{R}$ and satisfy the following equation:
$c^2d^2-2bcde+b^2e^2-2acdf+a^2f^2-2abef=0$
Show that the above equation cannot hold if all of the unknown quantities
$a,b,c,d,e,f$ are nonzero.
I did the following: I rewrote the above equation in 3 equivalent ways:
$(cd-be)^2+(af)^2=2af(cd+be)$
$(af-be)^2+(cd)^2=2cd(af+be)$
$(cd-af)^2+(be)^2=2be(cd+af)$
Since the L.H.S. of the three equations is non-negative, we have
$(af)(cd+be)\geq 0$
$(cd)(af+be)\geq 0$
$(be)(af+cd)\geq 0$
I'm stuck from here. Any help would be appreciated.