The one-point 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 E0 and P to X(1,0).
Proof
By Theorem AC2, there is an affine transformation t
which maps E to E0
Suppose that t maps P to P'. As P is on E, P' is on E0.
We can map E0 to E0 by any element of E0(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 E0, and P to X.
If s maps E to E0, and P to X,
then so(rot)-1 maps E0
to E0, and X to X,
so the composite is the identity or r0 reflection in the x-axis. It is then easy
to see that s = rot or r0orot =
ρot.
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