proof of the affine tangent theorem

The Tangent Theorem
If P is a point on the conic C, then there is a unique line L tangent to C at P.

Let C0 be the standard conic affine congruent to C, and let T be a point where C0 cuts
its main (euclidean) axis of symmetry. The conic C0 has a unique tangent L at T.

By the affine transitivity theorem, there is an affine transformation t which maps
C to C0, and P to T. By the affine tangency theorem, applied to t-1, the line t-1(L)
is the tangent to C at P.

If M is any tangent at P, t(M) is a tangent to C0 at T, so t(M) = L, i.e. M = t-1(L).

affine tangents page