Similar triangles

Definition

Let $ABC$ and $A'B'C'$ be triangles in the Euclidean space $\mathbb{R}^n$, and let $d$ be the Euclidean distance. The triangles are similar, written $ABC \sim A'B'C'$, if there is a real number $r > 0$ such that $$d(A', B') = r \, d(A, B), \quad d(B', C') = r \, d(B, C) \quad \text{and} \quad d(C', A') = r \, d(C, A).$$ The number $r$ is the ratio of the similarity.