In fact, these groups are related to the hyperbolic group. This is done in the models page . Here, we prove a stronger form of the theorem that all conics are projective congruent. This has the advantage that it is symmetric in x, y and z. If P,Q and R are p-points on the projective conic C , then there is a unique projective transformation which maps C to C 0 and maps the points P,Q,R to X,Y,Z respectively.
All conics are projective congruent.
Z. We simly apply the map v to get a figure in which the conic is the standard conic, do any necessary calculations for this, and then recover the result in general by applying v -1 . If A,B,C are p-points such that AB touches C at P, BC touches C at R and CA touches C at Q, then AR, BQ and CP are concurrent. If the problem involves further points on the conic, the following result allows us to describe the corresponding points on the standard conic. A similar result holds for any conic C . If A,B,C.A',B',C' are p-points on a conic C , and AB', A'B meet at P BC',B'C at Q, and CA',C'A at R, then P,Q,R are collinear.