**Euclid's Elements: Book I: Proposition 38**

**Proposition 38: Triangles which are on equal bases and in the same parallels are equal to one another.**
Let ABC, DEF be triangles on equal bases BC, EF and in the same parallels BF, AD; I say that the triangle ABC is equal to the triangle DEF.

For let AD be produced in both directions to G, H; through B let BG be drawn parallel to CA, I. 31 and through F let FH be drawn parallel to DE.

Then each of the figures GBCA, DEFH is a parallelogram; and GBCA is equal to DEFH; for they are on equal bases BC, EF and in the same parallels BF, GH. I. 36

Moreover the triangle ABC is half of the parallelogram GBCA; for the diameter AB bisects it. I. 34

And the triangle FED is half of the parallelogram DEFH; for the diameter DF bisects it. I. 34

(*But the halves of equal things are equal to one another.*)

Therefore the triangle ABC is equal to the triangle DEF.

Therefore etc.

Q.E.D.