The onepoint theorem for affine ellipses
If P is a point on an affine ellipse E, then there are exactly two affine transformations
which map E to E_{0} and P to X(1,0).
Proof
By Theorem AC2, there is an affine transformation t
which maps E to E_{0}
Suppose that t maps P to P'. As P is on E, P' is on E_{0}.
We can map E_{0} to E_{0} by any element of E_{0}(2).
There are two elements which
map P' to X, namely the rotation ρ about O through angle XOP', and the reflection
r in the bisector of angle XOP'. Then ρot and rot each have the
required property  mapping E to E_{0}, and P to X.
If s maps E to E_{0}, and P to X,
then so(rot)^{1} maps E_{0}
to E_{0}, and X to X,
so the composite is the identity or r_{0} reflection in the xaxis. It is then easy
to see that s = rot or r_{0}orot =
ρot.

