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.

Remarks

Similar triangles have equal corresponding angles. Conversely, the angle-angle criterion says that two pairs of equal angles are enough for similarity. Congruent triangles are the similar triangles with the ratio $r = 1$. Two triangles are similar if and only if a homothety followed by an isometry sends one onto the other, vertex by vertex. Any two equilateral triangles are similar.

Explanations

L1 human

A photo of a triangle, enlarged or reduced, gives a similar triangle. All sides change by the same factor, and the angles stay the same. For example, the triangles with sides $3, 4, 5$ and $6, 8, 10$ are similar with the ratio $2$.