Fig. 34
125. Desargues's theorem concerning conics through four points. Let DD' be any pair of points in the involution determined as above, and consider the conic passing through the five points K, L, M, N, D. We shall use Pascal's theorem to show that this conic also passes through D'. The point D' is determined as follows: Fix L and M as before (Fig. 34) and join D to L, giving on MN the point N'. Join N' to B, giving on LK the point K'. Then MK' determines the point D' on the line AA', given by the complete quadrangle K', L, M, N'. Consider the following six points, numbering them in order: D = 1, D' = 2, M = 3, N = 4, K = 5, and L = 6. We have the following intersections: B = (12-45), K' = (23-56), N' = (34-61); and since by construction B, N, and K' are on a straight line, it follows from the converse of Pascal's theorem, which is easily established, that the six points are on a conic. We have, then, the beautiful theorem due to Desargues:
The system of conics through four points meets any line in the plane in pairs of points in involution.
126. It appears also that the six points in involution determined by the quadrangle through the four fixed [pg 75] points belong also to the same involution with the points cut out by the system of conics, as indeed we might infer from the fact that the three pairs of opposite sides of the quadrangle may be considered as degenerate conics of the system.
127. Conics through four points touching a given line. It is further evident that the involution determined on a line by the system of conics will have a double-point where a conic of the system is tangent to the line. We may therefore infer the theorem
Through four fixed points in the plane two conics or none may be drawn tangent to any given line.