# Equiangular Triangles are Similar

## Theorem

Let two triangles have the same corresponding angles.

Then their corresponding sides are proportional.

Thus, by definition, such triangles are similar.

In the words of Euclid:

*In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles.*

(*The Elements*: Book $\text{VI}$: Proposition $4$)

## Proof

Let $\triangle ABC, \triangle DCE$ be equiangular triangles such that:

- $\angle ABC = \angle DCE$
- $\angle BAC = \angle CDE$
- $\angle ACB = \angle CED$

Let $BC$ be placed in a straight line with $CE$.

From Two Angles of Triangle are Less than Two Right Angles $\angle ABC + \angle ACB$ is less than two right angles.

As $\angle ACB = \angle DEC$, it follows that $\angle ABC + \angle DEC$ is also less than two right angles.

So from the Parallel Postulate, $BA$ and $ED$, when produced, will meet.

Let this happen at $F$.

We have that $\angle ABC = \angle DCE$.

So from Equal Corresponding Angles implies Parallel Lines:

- $BF \parallel CD$

Again, we have that $\angle ACB = \angle CED$.

Again from Equal Corresponding Angles implies Parallel Lines:

- $AC \parallel FE$

Therefore by definition $\Box FACD$ is a parallelogram.

Therefore from Opposite Sides and Angles of Parallelogram are Equal $FA = DC$ and $AC = FD$.

Since $AC \parallel FE$, it follows from Parallel Transversal Theorem that:

- $BA : AF = BC : CE$

But $AF = CD$ so:

- $BA : AF = BC : CE$

From Proportional Magnitudes are Proportional Alternately:

- $AB : BC = DC : CE$

Since $CD \parallel BF$, from Parallel Transversal Theorem:

- $BC : CE = FD : DE$

But $FD = AC$ so $BC : CE = AC : DE$.

So from Proportional Magnitudes are Proportional Alternately, $BC : CA = CE : ED$.

It then follows from Equality of Ratios Ex Aequali that $BA : AC = CD : DE$.

$\blacksquare$

## Historical Note

This proof is Proposition $4$ of Book $\text{VI}$ of Euclid's *The Elements*.

## Sources

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