The Tangent Theorem
If P is a point on the conic C, then there is a unique line L tangent to C at P.
Proof
Let C_{0} be the standard conic affine congruent to C,
and let T be a point where C_{0} cuts
its main (euclidean) axis of symmetry. The conic C_{0}
has a unique tangent L at T.
By the affine transitivity theorem, there is an affine
transformation t which maps
C to C_{0}, 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 C_{0} at T, so
t(M) = L, i.e. M = t^{1}(L).

