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 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". The lemma establishes an important property for