Similarity of Polygons is Equivalence

Theorem

Let $A, B, C$ be polygons.

If $A$ and $B$ are both similar to $C$, then $A$ is similar to $B$.

As Euclid defined it:

Figures which are similar to the same rectilineal figure are also similar to one another.

(The Elements: Book VI: Proposition $21$)

It is also worth noting that:

$A$ is similar to $A$, and so similarity between polygons is reflexive.

If $A$ is similar to $B$, then $B$ is similar to $A$, and so similarity between polygons is symmetric.

Hence the relation of similarity between polygons is an equivalence relation.

Proof

We have that $A$ is similar to $C$.

From Book VI Definition 1: Similar Rectilineal Figures, it is equiangular with it and the sides about the equal angles are proportional.

We also have that $B$ is similar to $C$.

Again, from Book VI Definition 1: Similar Rectilineal Figures, it is equiangular with it and the sides about the equal angles are proportional.

So by definition $A$ is similar to $B$.

The statements of reflexivity and symmetry are shown similarly.

It follows that if $A$ is similar to $B$, and $B$ is similar to $C$, then $A$ is similar to $C$.

Thus similarity between polygons is transitive.

Hence the result, by definition of equivalence relation.

$\blacksquare$

Historical Note

This is Proposition 21 of Book VI of Euclid's The Elements.

Euclid himself did not have the concept of an equivalence relation. However, the extra statements leading to the main result are sufficiently straightforward to justify adding the full proof here.