Euclid's Elements Book I

1. Let ABC, DEF be triangles on the equal bases BC, EF and in the same parallels BF, AD. The claim: the triangle ABC is equal to the triangle DEF. (Try dragging the points.)
2. Let AD be produced in both directions to G, H; through B let BG be drawn parallel to CA, and through F let FH be drawn parallel to DE.

   Postulate 2Proposition I.31

   Postulate 2: To produce a finite straight line continuously in a straight line.

   Proposition I.31: Through a given point to draw a straight line parallel to a given straight line.
3. Then each of the figures GBCA, DEFH is a parallelogram; and they are on the equal bases BC, EF and in the same parallels BF, GH; therefore they are equal.

   Proposition I.36

   Proposition I.36: Parallelograms which are on equal bases and in the same parallels are equal to one another.
4. The diameter AB bisects GBCA, so the triangle ABC is half of GBCA; and the diameter DF bisects DEFH, so the triangle FED is half of DEFH.

   Proposition I.34

   Proposition I.34: In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.
5. But the halves of equal things are equal to one another; therefore the triangle ABC is equal to the triangle DEF.

Being what it was required to prove.