 Gergonne's insight was that if a line L1 could be constructed such that A1 and B1 were guaranteed to fall on it, those two points could be identified as the intersection points of L1 with the given circle C1 (Figure 6). The remaining four tangent points would be located similarly, by finding lines L2 and L3 that contained A2 and B2, and A3 and B3, respectively. To construct a line such as L1, two points must be identified that lie on it; but these points need not be the tangent points. Gergonne was able to identify two other points for each of the three lines. One of the two points has already been identified: the radical center G lies on all three lines (Figure 6).
