Question 888022
<pre>
{{{z[1]+z[2]+z[3]}}}{{{""=""}}}{{{"0"}}}

{{{z[3]}}}{{{""=""}}}{{{-z[2]-z[1]}}}

Substitute in

{{{z[1]^2+z[2]^2+z[3]^2}}}{{{""=""}}}{{{"0"}}}

{{{z[1]^2+z[2]^2+(-z[2]-z[1])^2}}}{{{""=""}}}{{{"0"}}}

{{{z[1]^2+z[2]^2+(z[2]^2+2z[1]z[2]+z[1]^2)}}}{{{""=""}}}{{{"0"}}}

{{{z[1]^2+z[2]^2+z[2]^2+2z[1]z[2]+z[1]^2}}}{{{""=""}}}{{{"0"}}}

{{{2z[1]^2+2z[2]^2+2z[1]z[2]}}}{{{""=""}}}{{{"0"}}}

{{{z[1]^2+z[2]^2+z[1]z[2]}}}{{{""=""}}}{{{"0"}}}

{{{z[2]^2+z[1]z[2]+z[1]^2}}}{{{""=""}}}{{{"0"}}}

{{{z[2]}}}{{{""=""}}}{{{(-b +- sqrt( b^2-4*a*c ))/(2*a) }}}

{{{z[2]}}}{{{""=""}}}{{{(-z[1] +- sqrt(z[1]^2-4*1*z[1]^2 ))/(2*1) }}}


{{{z[2]}}}{{{""=""}}}{{{(-z[1] +- sqrt(z[1]^2-4z[1]^2 ))/2 }}}

{{{z[2]}}}{{{""=""}}}{{{(-z[1] +- sqrt(-3z[1]^2))/2 }}}

{{{z[2]}}}{{{""=""}}}{{{(-z[1] +- z[1]*i*sqrt(3))/2 }}}

{{{z[2]}}}{{{""=""}}}{{{z[1](-1 +- i*sqrt(3))/2 }}}

{{{abs(z[2])}}}{{{""=""}}}{{{ abs(z[1]) * abs( (-1 +- i*sqrt(3))/2) ) }}}


{{{abs(z[2])}}}{{{""=""}}}{{{ abs(z[1]) * abs( -1/2 +- i*sqrt(3)/2 ) }}}

{{{abs(z[2])}}}{{{""=""}}}{{{ abs(z[1]) * sqrt((-1/2)^2 +("" +- sqrt(3)/2)^2 ) }}}

{{{abs(z[2])}}}{{{""=""}}}{{{ abs(z[1]) * sqrt(1/4 +3/4 ) }}}

{{{abs(z[2])}}}{{{""=""}}}{{{ abs(z[1]) * sqrt(1 ) }}}

{{{abs(z[2])}}}{{{""=""}}}{{{ abs(z[1]) *(1 ) }}}

{{{abs(z[2])}}}{{{""=""}}}{{{ abs(z[1]) }}}

by symmetry, we may swap {{{z[2]}}} and {{{z[3]}}}

{{{abs(z[3])}}}{{{""=""}}}{{{ abs(z[1]) }}}

{{{abs(z[1])}}}{{{""=""}}}{{{ abs(z[2]) }}}{{{""=""}}}{{{ abs(z[3]) }}}

Edwin</pre>