Euclid's Elements Book I

1. Let ABC, DBC be triangles on the same base BC and in the same parallels AD, BC. The claim: the triangle ABC is equal to the triangle DBC. (Try dragging A, B, C or D.)
2. Let AD be produced in both directions to E, F; through B let BE be drawn parallel to CA, and through C let CF be drawn parallel to BD.

   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 EBCA, DBCF is a parallelogram; and they are on the same base BC and in the same parallels BC, EF; therefore they are equal.

   Proposition I.35

   Proposition I.35: Parallelograms which are on the same base and in the same parallels are equal to one another.
4. The diameter AB bisects the parallelogram EBCA, so the triangle ABC is half of EBCA; and the diameter DC bisects DBCF, so the triangle DBC is half of DBCF.

   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 DBC.

Being what it was required to prove.