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
A Perfect Circle two became friends. Three years later, Keenan offered Howerdel, who was looking for lodging, a room in his North Hollywood home. This provided Howerdel the opportunity to play demos of his music for Keenan. Pleased with what he heard, Keenan remarked, "I can hear myself singing [those songs]." Although he originally desired a female vocalist, Howerdel agreed that Keenan would be a good fit, and A Perfect Circle was formed a short time later. The two rounded out the band's initial lineup with bassist and violinist Paz Lenchantin, former Failure guitarist Troy Van Leeuwen, and Primus drummer Tim Alexander. The