Triangles which are on equal bases and in the same parallels equal one another.

Let *ABC* and *DEF* be triangles on equal bases *BC* and *EF* and in the same parallels *BF* and *AD.*

I say that the triangle *ABC* equals the triangle *DEF.*

Produce *AD* in both directions to *G* and *H.* Draw *BG* through *B* parallel to *CA,* and draw *FH* through *F* parallel to *DE.*

Then each of the figures *GBCA* and *DEFH* is a parallelogram, and *GBCA* equals *DEFH,* for they are on equal bases *BC* and *EF* and in the same parallels *BF* and *GH.*

Moreover the triangle *ABC* is half of the parallelogram *GBCA,* for the diameter *AB* bisects it. And the triangle *FED* is half of the parallelogram *DEFH,* for the diameter *DF* bisects it.

Therefore the triangle *ABC* equals the triangle *DEF.*

Therefore *triangles which are on equal bases and in the same parallels equal one another.*

Q.E.D.