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 $B$.


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

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

 * Euclid-VI-21.png

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

From, 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, 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.