triangles and parallels

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

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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 etc.

## 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