Properties of Homothety
What is a Homothety?
A homothety (also called a dilation or homothecy) is a geometric transformation that enlarges or reduces a figure relative to a fixed point, called the center of homothety.
Formal Definition: A homothety with center $O$ and ratio $k$ (where $k \neq 0$) is a transformation $H_{O,k}$ that maps each point $P$ to a point $P'$ such that: $$\overrightarrow{OP'} = k \cdot \overrightarrow{OP}$$
The number $k$ is called the ratio of homothety (or scale factor).
- If $|k| > 1$: the figure is enlarged
- If $0 < |k| < 1$: the figure is reduced
- If $k > 0$: the image is on the same side of $O$ as the original
- If $k < 0$: the image is on the opposite side of $O$ (includes a point reflection)
- If $k = 1$: the transformation is the identity
- If $k = -1$: the transformation is a point reflection through $O$
🔍 Interactive Exploration: Homothety in Action
Drag the slider to change the ratio $k$. Drag the center point $O$ to move the center of homothety. Observe how the triangle transforms!
Interactive visualization — available in the full lesson