**Euclid's Elements: Book I: Proposition 40**

**Proposition 40: Equal triangles which are on equal bases and on the same side are also in the same parallels.**
Let ABC, CDE be equal triangles on equal bases BC, CE and on the same side.

I say that they are also in the same parallels.

For let AD be joined; I say that AD is parallel to BE.

For, if not, let AF be drawn through A parallel to BE I. 31, and let FE be joined.

Therefore the triangle ABC is equal to the triangle FCE; for they are on equal bases BC, CE and in the same parallels BE, AF. I. 38

But the triangle ABC is equal to the triangle DCE; therefore the triangle DCE is also equal to the triangle FCE, C.N. 1 the greater to the less: which is impossible. Therefore AF is not parallel to BE.

Similarly we can prove that neither is any other straight line except AD; therefore AD is parallel to BE.

Therefore etc.

Q.E.D.