Congruent triangles

Definition

Let $n$ be a positive integer, and let $ABC$ and $A'B'C'$ be triangles in the Euclidean space $\mathbb{R}^n$. The triangles are congruent, written $ABC \cong A'B'C'$, if there is an isometry $f : \mathbb{R}^n \to \mathbb{R}^n$ with $f(A) = A'$, $f(B) = B'$ and $f(C) = C'$. An isometry is a map $f$ with $d(f(x), f(y)) = d(x, y)$ for all points $x$ and $y$, where $d$ is the Euclidean distance.

Remarks

Congruence is an equivalence relation on triangles. Congruent triangles have equal corresponding sides and equal corresponding angles. The side-side-side criterion gives the converse. The definition depends on the order of the vertices. For example, $ABC \cong ACB$ holds only when $d(A, B) = d(A, C)$. Mathlib's Congruent asks for equal distances between all pairs of corresponding points of two indexed families. In a Euclidean space this is equivalent, by the side-side-side criterion.

Explanations

L1 human

Two triangles are congruent if you can cut one out of paper and lay it exactly on the other. You may slide it, turn it, and flip it over. Then corresponding sides and corresponding angles match.