Euclid's Elements Book I

1. Let ABC, DBC be equal triangles on the same base BC and on the same side of it. The claim: they are also in the same parallels, that is, AD is parallel to BC. (Try dragging A, B, C or D.)
2. If not, through the point A let AE be drawn parallel to the straight line BC, and let EC be joined. (E is drawn as this assumption would place it.)

   Proposition I.31Postulate 1

   Proposition I.31: Through a given point to draw a straight line parallel to a given straight line.

   Postulate 1: To draw a straight line from any point to any point.
3. Then the triangle ABC is equal to the triangle EBC, for it is on the same base BC with it and in the same parallels.

   Proposition I.37

   Proposition I.37: Triangles which are on the same base and in the same parallels are equal to one another.
4. But ABC is equal to DBC; therefore DBC is also equal to EBC, the greater to the less, which is impossible.

   Common Notion 1Common Notion 5

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

   Common Notion 5: The whole is greater than the part.
5. Therefore AE is not parallel to BC; and similarly no other straight line except AD is. Therefore AD is parallel to BC.

Being what it was required to prove.