triangles in parallels equality
===============================

.. index:: proof, triangles, parallels

.. image:: elem.1.prop.37.png
   :align: right
   :width: 300px

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

===

Let `ABC`, `DBC` be triangles on the same base `BC` and in the same parallels `AD`, `BC`; I say that the triangle `ABC` is equal to the triangle `DBC`.

Let `AD` be produced in both directions to `E`, `F`; through `B` let `BE` be drawn parallel to `CA`, [I.31] and through `C` let `CF` be drawn parallel to `BD`. [I.31] 

Then each of the figures `EBCA`, `DBCF` is a parallelogram; and they are equal, 

for they are on the same base `BC` and in the same parallels `BC`, `EF`. [I.35]

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

And the triangle `DBC` is half of the parallelogram `DBCF`; for the diameter `DC` bisects it. [I.34]

[But the halves of equal things are equal to one another.] [^I.37:1]

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

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.35]: /elem.1.35 "Book 1 - Proposition 35"

## Footnotes

[^I.37:1]: But the halves of equal things are equal to one another
    Here and in the next proposition Heiberg brackets the words <quote>But the halves of equal things are equal to one another</quote> on the ground that, since the <title>Common Notion</title> which asserted this fact was interpolated at a very early date (before the time of Theon), it is probable that the words here were interpolated at the same time. Cf. note above (p. 224) on the interpolated <title>Common Notion</title>.