Euclid's Elements Book I

1. Let ABC, CDE be equal triangles on the equal bases BC, CE and on the same side. The claim: they are also in the same parallels, that is, AD is parallel to BE. (Try dragging A, B, C or D.)
2. If not, through A let AF be drawn parallel to BE, and let FE be joined. (F 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 FCE, for they are on the equal bases BC, CE and in the same parallels BE, AF.

   Proposition I.38

   Proposition I.38: Triangles which are on equal bases and in the same parallels are equal to one another.
4. But ABC is equal to DCE; therefore DCE is also equal to FCE, 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 AF is not parallel to BE; and similarly no other straight line except AD is. Therefore AD is parallel to BE.

Being what it was required to prove.