---
tag: 01PNAVFMYZ
kind: definition
name: Similar triangles
msc: [51M04]
defines: 0YF6QXXWB5
uses: [0CTNFGYMM2, 0NPPGHQNXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Similarity_(geometry)
    locator: Similar triangles
---

## Definition

Let $ABC$ and $A'B'C'$ be [[triangles|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].
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|0NZVJYZVTZ]] says that two pairs of equal angles are enough for similarity.
[[Congruent triangles|093EDENVA1]] are the similar triangles with the ratio $r = 1$.
Two triangles are similar if and only if a [[homothety|0ZG1PNEP35]] 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 $2$.