the three point theorem

 In projective geometry, all conics are congruent, so the projective symmetry groups are conjugate subgroups of P(2). 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 is the analogue of the one- and two-point theorems for affine geometry. We will choose as our standard projective conic the conic C0 : xy+yz+xz = 0. This has the advantage that it is symmetric in x, y and z. It also contains the standard points X[1,0,0], Y[0,1,0] and Z[0,0,1], but not U[1,1,1]. The Three Point Theorem If P,Q and R are p-points on the projective conic C, then there is a unique projective transformation which maps C to C0 and maps the points P,Q,R to X,Y,Z respectively. Of course, since all conics are projective congruent to C0, we have the Theorem All conics are projective congruent. This result has the consequence that any theorem in projective geometry involving three (or more) points on a conic may be reduced to a problem about the standard conic and the points X,Y,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. Some examples : The Three Tangents Theorem Suppose that C is a conic. 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. The Inscribed Quadrilateral Theorem If A,B,C,D are distinct p-points on a p-conic C, and AB meets CD in P, BC meets AD in Q, and CA meets BD in R, then QR is the polar of P with respect to C. Note that this allows us to draw the polar of a point P not on a conic C : Draw two chords AB and CD through P Find the intersections Q of BC, AD, and R of CA, BD. The required polar is QR. If the problem involves further points on the conic, the following result allows us to describe the corresponding points on the standard conic. The Parametrisation Theorem If the p-point P lies on the conic C0 : xy+yz+zx = 0, and P ≠ Z, then P has the form (t,1-t,t(t-1)) for some real t. A similar result holds for any conic C. We know that C is projective congruent to C0, so there is a projective transformation t with C = t(C0). Suppose that t has matrix M. Each p-point on C has the form t(P) for some P on C0. If P ≠ Z, then P =[t,1-t,t(t-1)] for some t, and t(P) has the form M(t,1-t,t(t-1))T. Thus, t(P) has the form (f(t),g(t),h(t)), where f,g,h are quadratic in t. This leads to the description of the symmetry group of a projective conic. Pascal's Theorem 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. A related result, useful in proving other theorems is the five point theorem.