triangle and parallels equality
===============================

.. index:: proof, triangles, parallels

Triangles which are on equal bases and in the same parallels are equal to one another.

===

Let `ABC`, `DEF` be triangles on equal bases `BC`, `EF` and in the same parallels `BF`, `AD`; I say that the triangle `ABC` is equal to the triangle `DEF`. 


For let `AD` be produced in both directions to `G`, `H`; through `B` let `BG` be drawn parallel to `CA`, [I.31] and through `F` let `FH` be drawn parallel to `DE`.


Then each of the figures `GBCA`, `DEFH` is a parallelogram; and `GBCA` is equal to `DEFH`; <pb n="334"/>


for they are on equal bases `BC`, `EF` and in the same parallels `BF`, `GH`. [I.36]


Moreover the triangle `ABC` is half of the parallelogram `GBCA`; for the diameter `AB` bisects it. [I.34]


And the triangle `FED` is half of the parallelogram `DEFH`; for the diameter `DF` bisects it. [I.34]


[But the halves of equal things are equal to one another.]


Therefore the triangle `ABC` is equal to the triangle `DEF`.


Therefore etc.

- Q. E. D.

## References


[I.31]: /elem.1.31 "Book 1 - Proposition 31"
[I.34]: /elem.1.34 "Book 1 - Proposition 34"
[I.36]: /elem.1.36 "Book 1 - Proposition 36"