Dividing A Circle Into Areas - Lemma

Lemma

If we already have n points on the circle and add one more point, we draw n lines from the new point to previously existing points. Two cases are possible. In the first case (a), the new line passes through a point where two or more old lines (between previously existing points) cross. In the second case (b), the new line crosses each of the old lines in a different point. It will be useful to know the following fact.

Lemma. We can choose the new point A so that case b occurs for each of the new lines.

Proof. Notice that, for the case a, three points must be on one line: the new point A, the old point O to which we draw the line, and the point I where two of the old lines intersect. Notice that there are n old points O, and hence finitely many points I where two of the old lines intersect. For each O and I, the line OI crosses the circle in one point other than O. Since the circle has infinitely many points, it has a point A which will be on none of the lines OI. Then, for this point A and all of the old points O, case b will be true.

This lemma means that, if there are k lines crossing AO, then each of them crosses AO at a different point and k+1 new areas are created by the line AO.