Euclid's Elements Book I

1. Let ABCD, EFGH be parallelograms on the equal bases BC, FG and in the same parallels AH, BG. The claim: the parallelogram ABCD is equal to the parallelogram EFGH. (Try dragging the points.)
2. Let BE, CH be joined.

   Postulate 1

   Postulate 1: To draw a straight line from any point to any point.
3. Since BC is equal to FG, and FG is equal to EH, BC is also equal to EH; and they are parallel.

   Proposition I.34Common Notion 1

   Proposition I.34: In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.

   Common Notion 1: Things which are equal to the same thing are also equal to one another.
4. So EB, HC, which join them at their ends, are also equal and parallel; therefore EBCH is a parallelogram.

   Proposition I.33

   Proposition I.33: The straight lines joining equal and parallel straight lines at the extremities which are in the same directions respectively are themselves also equal and parallel.
5. The parallelogram EBCH is on the same base BC as ABCD and in the same parallels; therefore EBCH is equal to ABCD.

   Proposition I.35

   Proposition I.35: Parallelograms which are on the same base and in the same parallels are equal to one another.
6. For the same reason, EBCH is on the same base EH as EFGH; therefore EBCH is equal to EFGH.

   Proposition I.35

   Proposition I.35: Parallelograms which are on the same base and in the same parallels are equal to one another.
7. Therefore the parallelogram ABCD is equal to the parallelogram EFGH.

   Common Notion 1

   Common Notion 1: Things which are equal to the same thing are also equal to one another.

Being what it was required to prove.