Then, since the straight line AC falls upon the parallels AD, EC,
the angle ACE is equal to the angle CAD. [I. 29]
But the angle CAD is by hypothesis equal to the angle BAD;
therefore the angle BAD is also equal to the angle ACE.
Again, since the straight line BAE falls upon the parallels AD, EC,
the exterior angle BAD is equal to the interior angle AEC. [I. 29]
But the angle ACE was also proved equal to the angle BAD;
therefore the angle ACE is also equal to the angle AEC,
so that the side AE is also equal to the side AC. [I. 6]
And, since AD has been drawn parallel to EC, one of the sides of the triangle BCE,
therefore, proportionally, as BD is to DC, so is BA to AE.
But AE is equal to AC; [VI. 2]
therefore, as BD is to DC, so is BA to AC.