In equiangular triangles the sides about the equal angles are proportional where the corresponding sides are opposite the equal angles.

Let *ABC* and *DCE* be equiangular triangles having the angle *ABC* equal to the angle *DCE,* the angle *BAC* equal to the angle *CDE,* and the angle *ACB* equal to the angle *CED.*

I say that in the triangles *ABC* and *DEC* the sides about the equal angles are proportional where the corresponding sides are opposite the equal angles.

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

Then, since the sum of the angles *ABC* and *ACB* is less than two right angles, and the angle *ACB* equals the angle *DEC,* therefore the sum of the angles *ABC* and *DEC* is less than two right angles. Therefore *BA* and *ED,* when produced, will meet. Let them be produced and meet at *F.*

Now, since the angle *DCE* equals the angle *ABC, DC* is parallel to *FB.* Again, since the angle *ACB* equals the angle *DEC, AC* is parallel to *FE.*

Therefore *FACD* is a parallelogram, therefore *FA* equals *DC,* and *AC* equals *FD.*

And, since *AC* is parallel to a side *FE* of the triangle *FBE,* therefore *BA* is to *AF* as *BC* is to *CE.*

But *FD* equals *AC,* therefore *BC* is to *CE* as *AC* is to *DE,* and alternately *BC* is to *CA* as *CE* is to *ED.*

Since then it was proved that *AB* is to *BC* as *DC* is to *CE,* and *BC* is to *CA* as *CE* is to *ED,* therefore, *ex aequali, BA* is to *AC* as *CD* is to *DE.*

Therefore, *in equiangular triangles the sides about the equal angles are proportional where the corresponding sides are opposite the equal angles.*

Q.E.D.

Euclid has placed the triangles in particular positions in order to employ this particular proof. Such positioning is common in Book VI and is easily justified.

This proposition implies that equiangular triangles are similar, a fact proved in detail in the proof of proposition VI.8. It also implies that triangles similar to the same triangle are similar to each other, also proved in detail in VI.8. The latter statement is generalized in VI.21 to rectilinear figures in general.

This proposition is frequently used in the rest of Book VI starting with the next proposition, its converse. It is also used in Books X through XIII.