Similar triangles and homothety (constructions)

Triangles $ABC$ and $A'B'C'$ are similar if $\angle A = \angle A'$, $\angle B = \angle B'$, $\angle C = \angle C'$, and the ratios of their corresponding sides are equal: $\frac{AB}{A'B'} = \frac{BC}{B'C'} = \frac{CA}{C'A'}$. Homothety with center $O$ and ratio $k$ maps each point $P$ to a point $P'$ such that $\overrightarrow{OP'} = k \cdot \overrightarrow{OP}$.

Important properties

• Similar triangles have equal corresponding angles and proportional sides.

• Homothety preserves angles and maps lines to parallel lines.

• A homothety with ratio $k > 1$ enlarges, $0 < k < 1$ reduces, and $k < 0$ reflects and scales.

• Any two similar triangles can be mapped onto each other by a homothety (possibly followed by a translation or rotation).