If a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle.

Let the parallelogram *ABCD* have the same base *BC* with the triangle *EBC,* and let it be in the same parallels *BC* and *AE.*

I say that the parallelogram *ABCD* is double the triangle *BEC.*

Join *AC.*

Then the triangle *ABC* equals the triangle *EBC,* for it is on the same base *BC* with it and in the same parallels *BC* and *AE.*

But the parallelogram *ABCD* is double the triangle *ABC,* for the diameter *AC* bisects it, so that the parallelogram *ABCD* is also double the triangle *EBC.*

Therefore *if a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle.*

Q.E.D.