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