Homothety (central similarity)

Dependencies: SAS similarity axiom (two sides in proportion + included angle equal ⇒ similar), Corresponding/alternate angles ⇔ lines parallel (corresponding angles equal ⇒ parallel).

Statement

Fix a point $O$ in the plane (the center of homothety) and a nonzero real number $k$ (the ratio of homothety). The homothety with center $O$ and ratio $k$ is the map sending each point $P$ to $P'$ defined by

It is a similarity transformation: when it sends any pair of points $P,Q$ simultaneously to $P',Q'$,

and $H_{O,\,k}$ is angle-preserving — $\triangle OPQ \sim \triangle OP'Q'$ with similarity ratio $|k|$.