The conjecture can be proven like this: (This follows the work in my previous post.)
Choose 3 roots, p, q, and r. (For the moment I am considering them all to be real numbers, the proof for complex roots is similar.)
So the cubic is:
You wish to find the line tangent to this cubic at the point . It suffices to choose this point as any other choice of roots to average merely shuffles the p, q, r's around.
The slope of this line will be the first derivative of y evaluated at the average x value above.
Now find out what the y value of the point on the tangent line is. (Evaluate y for the average x value.)
Now you need the intercept of the tangent line. So y = mx + b, solve for b.
Now you need to show that this line intersects the point (r, 0). So solve 0 = mx + b for x. The solution should be x = r.