# Similarity of Polygons is Equivalence Relation

## Theorem

Let $A, B, C$ be polygons.

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

In the words of Euclid:

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

(*The Elements*: Book $\text{VI}$: 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 $\text{VI}$ 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 $\text{VI}$ 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 proof is Proposition $21$ of Book $\text{VI}$ 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.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{VI}$. Propositions