parallelograms and triangles

../../../../_images/elem.1.prop.41.png

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

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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.

## References

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