Question 1117926
Let the roots be {{{ w[k] }}} where k=0,1,2.<br>

The three roots can be found by  {{{ w^3 = e^(2(pi)ki) }}} for k=0,1,2 <br>

Using this property:  {{{ root(n,r*e^m) = (root(n,r))*matrix(3,3,"","","","",e^(m/n),"","","","") }}} <br>


{{{ w =  matrix(3,3,"","","","",e^(2(pi)ki/3),"","","","") }}} for k=0,1,2<br>

k=0:  {{{ w[0] = e^0 = 1 }}}
k=1:  {{{ w[1] = matrix(3,3,"","","","",e^(2(pi)*1*i/3), "","","","") = cos(2(pi)/3) + i*sin(2(pi)/3) =  -1/2 + i*sqrt(3)/2 }}}
k=2:  {{{ w[2] = matrix(3,3,"","","","",e^(2(pi)*2*i/3), "","","","") = cos(4(pi)/3) + i*sin(4(pi)/3) = -1/2 - i*sqrt(3)/2 }}} <br>

Now if you plot the real values on the x-axis and imaginary parts along the y-axis, and you connect each point to form a triangle, you will find the distance between each of the three points (vertices) is {{{ sqrt(3) }}} units.   Thus completing the proof.   There may be a more efficient method, but this method does work.