If you like that, then try this.

Take a row of pascal's triangle, add a zero in front (shift it to the right by one place), and divide it by the row below it.

    0 1  4  6 4 1 /
    1 5 10 10 5 1
    = ...

From this you can prove that a linear bezier line (of any degree) must have equally spaced control points.