Given any two million points, there are at most \(\binom{2\,000\,000}{2}\) lines that can be drawn that pass through two of these points.
Therefore, lines passing through two of the points have at most \(\binom{2\,000\,000}{2}\) different slopes.
Choose a slope, \(m\), that is not in this list of slopes.
Any line with slope \(m\) cannot pass through \(2\) or more of the points at the same time, so will pass through either \(0\) points or \(1\) point.
Draw a line with this slope with all of the points on one side.
Slide this line so that it starts to “pass” the points one by one.
Since the line passes \(1\) point at a time, then there will be an instant when there are \(999\,999\) points on one side, \(1\) point on the line, and \(1\,000\,000\) points on the other side.
Each of the \(1\,000\,000\) points on one side of the line has a fixed perpendicular distance to the line.
Choose a distance less than the smallest of these distances and slide the line by that distance. It will move off of the
\(1\) point and will not pass any more points.
Therefore, there are \(1\,000\,000\) points now on each side of the line, as required. \(\square\)