If an angle of a triangle is bisected by a straight line cutting the base, then the segments of the base have the same ratio as the remaining sides of the triangle; and, if segments of the base have the same ratio as the remaining sides of the triangle, then the straight line joining the vertex to the point of section bisects the angle of the triangle.

Let *ABC* be a triangle, and let the angle *BAC* be bisected by the straight line *AD.*

I say that *DB* is to *DC* as *AB* is to *AC.*

Draw *CE* through *C* parallel to *DA,* and carry *AB* through to meet it at *E.*

Then, since the straight line *AC* falls upon the parallels *AD* and *EC,* the angle *ACE* equals the angle *CAD.*

But the angle *CAD* equals the angle *BAD* by hypothesis, therefore the angle *BAD* also equals the angle *ACE.*

Again, since the straight line *BAE* falls upon the parallels *AD* and *EC,* the exterior angle *BAD* equals the interior angle *AEC.*

But the angle *ACE* was also proved equal to the angle *BAD,* therefore the angle *ACE* also equals the angle *AEC,* so that the side *AE* also equals the side *AC.*

And, since *AD* is parallel to *EC,* one of the sides of the triangle *BCE,* therefore, proportionally *DB* is to *DC* as *AB* is to *AE.*

But *AE* equals *AC,* therefore *DB* is to *DC* as *AB* is to *AC.*

Next, let *DB* be to *DC* as *AB* is to *AC.* Join *AD.*

I say that the straight line *AD* bisects the angle *BAC.*

With the same construction, since *DB* is to *DC* as *AB* is to *AC,* and also *DB* is to *DC* as *AB* is to *AE,* for *AD* is parallel to *EC,* one of the sides of the triangle *BCE,* therefore also *AB* is to *AC* as *AB* is to *AE.*

Therefore *AC* equals *AE,* so that the angle *AEC* also equals the angle *ACE.*

But the angle *AEC* equals the exterior angle *BAD,* and the angle *ACE* equals the alternate angle *CAD,* therefore the angle *BAD* also equals the angle *CAD.*

Therefore the straight line *AD* bisects the angle *BAC.*

Therefore, *if an angle of a triangle is bisected by a straight line cutting the base, then the segments of the base have the same ratio as the remaining sides of the triangle; and, if segments of the base have the same ratio as the remaining sides of the triangle, then the straight line joining the vertex to the point of section bisects the angle of the triangle.*

Q.E.D.