triangles and parallels
=======================

.. index:: proof, triangles, parallels

.. image:: elem.1.prop.40.png
   :align: right
   :width: 300px

Equal triangles which are on equal bases and on the same side are also in the same parallels

===

Let `ABC`, `CDE` be equal triangles on equal bases `BC`, `CE` and on the same side.

I say that they are also in the same parallels.

For let `AD` be joined; I say that `AD` is parallel to `BE`.

For, if not, let `AF` be drawn through `A` parallel to `BE` [I.31], and let `FE` be joined. 

Therefore the triangle `ABC` is equal to the triangle `FCE`; for they are on equal bases `BC`, `CE` and in the same parallels `BE`, `AF`. [I.38]

But the triangle `ABC` is equal to the triangle `DCE`; 

- therefore the triangle `DCE` is also equal to the triangle `FCE`, [I.c.n.1] the greater to the less: which is impossible. Therefore `AF` is not parallel to `BE`.

Similarly we can prove that neither is any other straight line except `AD`; 

- therefore `AD` is parallel to `BE`.

Therefore etc. 

- Q. E. D.

## References

[I.31]: /elem.1.31 "Book 1 - Proposition 31"
[I.38]: /elem.1.38 "Book 1 - Proposition 38"
[I.c.n.1]: /elem.1.c.n.1 "Book 1 - Common Notion 1"

## Footnotes