Homothety (GMT)

A homothety is a transformation of the plane that enlarges or shrinks figures from a fixed point, called the center, by a certain ratio (the scale factor). Every point $P$ is mapped to a point $P'$ such that the center $O$, $P$, and $P'$ are collinear, and $OP' = k imes OP$, where $k$ is the scale factor.

Important properties

• Homothety preserves the shape of figures but changes their size.

• Lines through the center of homothety remain unchanged.

• Parallel lines remain parallel after homothety.

• If $k > 0$, the image is on the same side of $O$ as the original point; if $k < 0$, it is on the opposite side.