Dividing a circle into areas "F" can also be derived, which will solve the problem. Its vertices include the "n" points on the circle, referred to as the exterior vertices, as well as the interior vertices, the intersections of distinct chords in the interior of the circle. The "generic intersection" assumption made above guarantees that each interior vertex is the intersection of no more than two chords. Thus the main task in determining "V" is finding the number of interior vertices. As a consequence of the lemma, any two intersecting chords will uniquely determine an interior vertex. These chords are in turn uniquely determined by
Dividing a circle into areas 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"