Homothety

A homothety with center $O$ and scale factor $k$ maps any point $P$ to a point $P'$ such that $O$, $P$, and $P'$ are collinear, and $OP' = k \cdot OP$. If $k > 0$, $P'$ is on the same side of $O$ as $P$; if $k < 0$, $P'$ is on the opposite side.

Important properties

• Homothety preserves the shape of figures (similarity).

• Lengths are multiplied by $|k|$; angles remain unchanged.

• Lines map to parallel lines (unless they pass through the center).

• The center of homothety is a fixed point.