proof of theorem AC2

 Theorem AC2 Each parabola is affine congruent to P0 : y2 = x. Each ellipse or circle is affine congruent to E0 : x2 + y2 = 0. Each hyperbola is affine congruent to H0 : xy  = 1. Proof (1) Suppose that P is a parabola. We already know that there is a similarity s mapping P to P0. As S(2) is a subgroup of A(2), s is an affine transformation, so the parabolas are affine congruent. (2) Suppose that E is an ellipse or circle. From euclidean geometry, we know that there is a euclidean (and hence affine) transfromation s such that s(E) has equation x2/a2+y2/b2 = 1, with a = b in the case of a circle. Now consider the transfromation r(x,y) = (x/a,y/b) - this has matrix diag(1/a,1/b), and hence is affine. It clearly maps the locus s(E) to the locus with equation x2+y2 = 1, i.e. to E0. Thus the affine transfromation ros maps E to E0, so the loci are affine congruent. (3) Suppose that H is a hyperbola. Much as in (2), there is an affine transformation t mapping H to the hyperbola E* : x2-y2 = 1. Now let u be the affine transformation u(x,y) = (x+y,x-y) - it has matrix with rows (1,1) and (1,-1). Then u(E*) has the equation xy = 1, i.e. is H0. Thus the affine transformation uot sends H to H0, so the hyperbolas are affine congruent.