**Euclid's Elements: Book I: Proposition 39**

**Proposition 39: Equal triangles which are on the same base and on the same side are also in the same parallels.**
Let ABC, DBC be equal triangles which are on the same base BC and on the same side of it;

(*I say that they are also in the same parallels.*)

And let AD be joined; I say that AD is parallel to BC.

For, if not, let AE be drawn through the point A parallel to the straight line BC, I. 31 and let EC be joined.

Therefore the triangle ABC is equal to the triangle EBC; for it is on the same base BC with it and in the same parallels. I. 37

But ABC is equal to DBC; therefore DBC is also equal to EBC, C.N. 1 the greater to the less: which is impossible.

Therefore AE is not parallel to BC.

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

Therefore etc.

Q.E.D.