proof of the affine tangencytheorem

The Affine Tangency Theorem
If L is a tangent at the point P on a conic C, and t is an affine transformation,
then t(L) is a tangent at t(P) on the conic t(C).

Since L is a tangent to C at P, we must have

  • LnC = {P}, and
  • there is a line L' parallel to L, with L'nC ={Q,R}, Q ≠ R.
Since t is a bijection, t(L)nt(C) = {t(P)}, and t(L')nt(C) = {t(Q),t(R)}.
As t is affine, t(L) is parallel to t(L'), so t(L) ia tangent to t(C) at t(P).

affine tangents page