proof of the three point theorem

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.

Proof
By the Fundamental Theorem of Projective Geometry, there is a projective
transformation t which maps P,Q,R to X,Y,Z. The Theorem applies since
the p-points P,Q,R lie on a conic, so are non-collinear.

Then X,Y,Z lie on C' = t(C).
Suppose that C' has equation ax2+bxy+cy2+fzx+gyz+hz2 = 0.
Since X is on C', a = 0. Similarly, as Y,Z lie on C', c = h = 0.
Thus C' has equation bxy+fzx+gyz = 0.

Suppose that b = 0. Then the equation factorizes as (fx+gy)z = 0, and hence
contains the p-line z = 1. This cannot occur as C, and C' are non-degenerate.
Thus, b ≠ 0. Similarly, f and g are non-zero. Now apply the transformation
u([x,y,z]) = [x/g,y/f,z/b]. This is projective since it is associated with the
matrix D = diag(1/g,1/f,1/b).

The equation of C' can be rewritten xy/gf+zx/bg+yz/gb = 0, so u(C') has
equation xy+zx+yz = 0, i.e. is C0.

Also, a simple calculation shows that u maps X to X, Y to Y and Z to Z,
so that v = uot maps C to C0, and P,Q,R to X,Y,Z.

Finally, suppose that w has the same effect on C,X,Y and Z. Then s = wov-1
maps C0 to itself and fixes X, Y and Z. If w has matrix M, then the effect on
X,Y,Z shows that M is diagonal. Say M = diag(p,q,r). Then a calculation shows
that sC0) has equation rxy+pyz+qzx = 0. But s maps C0 to C0, so r = p = q.
Thus, M is pI, so s is the identity transformation, i.e. w = v. Hence, v is unique.

three point theorem page