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. Lemma 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. proof (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.