parallelograms and triangles
============================

.. index:: proof, parallelograms, triangles

.. image:: elem.1.prop.41.png
   :align: right
   :width: 300px

If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle.

===

For let the parallelogram `ABCD` have the same base `BC` with the triangle `EBC`, and let it be in the same parallels `BC`, `AE`;

I say that the parallelogram `ABCD` is double of the triangle `BEC`.

For let `AC` be joined.

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 `BC`, `AE`. [I.37] 

But the parallelogram `ABCD` is double of the triangle `ABC`; 

- for the diameter `AC` bisects it; [I.34]

 so that the parallelogram `ABCD` is also double of the triangle `EBC`.

Therefore etc.

- Q. E. D.


## References

[I.34]: /elem.1.34 "Book 1 - Proposition 34"
[I.37]: /elem.1.37 "Book 1 - Proposition 37"