Number theory 5: Given $2n+2$ points in the plane......

Given $2n+2$ points in the plane, no three collinear,prove that two of them deter-mine a line that separates $n$ of the points from the other $n$ 

Solution 

Imagine the points lying on the map,and choose the westmost point, say $P_1$, as one of the two that will determine the line (there are at most two westmost point,choose any of them).
 Place a Cartesian system of coordinates with the origin at $P_1$,the x-axis in the direction west-esat, and the y-axis in the direction west-east, and the y-axis in the direction south-north. Order the rest of the points in an increasing sequence $P_2,P_3,...P_{2n+2}$ with respect to the oriented angles that $P_1P_{i}$ from with the x-axis.

This is possible because no three points are collinear and the angles are between $-90^{o}$ and $90^{o}$.
If we choose $P_1P_{n+2}$ to be the line, then $P_2,P_3,...P_{n+1}$ lie inside the angle formed by $P_1P_{n+2}$ and the negative half of the y-axis, and $P_{n+3},P_{n+4},...P_{2n+2}$ lie inside the angle formed by $P_1P_{n+2}$ and the positive half of y-axis,so the two sets of point are separated by the line $P_1P_{n+2}$, which show that $P_1$ and $P_{n+2}$ have the desired property.