proof of central conic theorem

The central conic theorem
If C is a central conic, with centre C, and t is an affine transformation,
then t(C) is central, with centre t(C).

We know that t(C) is of the same type as C, so is central.
If r is the half-turn with centre C, then r maps C to itself,
by the definition of centre.
By the affine half-turn theorem, torot-1 is the half-turn with
centre t(C). It clearly fixes t(C), so this has centre t(C).

central conics page