proof of the affine half-turn theorem

The affine half-turn theorem
If r is the half-turn with centre C, and t is any affine transformation,
then torot-1 is the half-turn with centre t(C).

To reduce the algebra required, we begin with a lemma about affine
transformations and their matrices.

If the affine transformations t and t' have matrices A and A' respectively, then
(1) t'ot has matrix A'A, and
(2) t-1 has matrix A-1.

(1) We have t(x) = Ax+b, and t'(x) = A'x+b' for some vectors b and b'. Then
t'ot(x) = A'(t(x))+b' = A'(Ax+b)+b' = A'Ax + (Ab+b'). Thus t'ot has matrix A'A.
(2) This follows from (1) since the identity map has matrix I, so t-1 has matrix B,
where BA = I, i.e. B = A-1.

proof of the theorem
Since r is a half-turn, it has matrix -I. Suppose that t has matrix A. By part (2)
of the Lemma, t-1 has matrix A-1. By part (1) of the Lemma, applied twice,
torot-1 has matrix A(-I)a-1 = -AA-1 = -I, so is a half-turn. Now, r fixes C, so
torot-1(t(C)) = tor(C) = t(C), i.e. torot-1 fixes t(C). Since a half-turn has a
unique fixed point, this is the centre of torot-1.

half-turns page