proofs of facts about half-turns facts about half-turns
(1) Suppose that the point P has image P' = r(P), and that these points have position vectors x and x', respectively. Then, as r is a half-turn about C, CP' = -CP, i.e. x' - c = -(x - c) = c - x, so that x' = 2c - x. (2) follows immediately from (1). (3) Supose that r and r' are the half-turns. then, by part (1), r(x) = b - x and r'(x) = b' - x, for some vectors b and b'. Composting these, we have ror'(x) = r(b' - x) = b - (b' - x) = x + (b - b'). Thus ror' is the translation by the vector (b - b'). |