.. index:: construction, midpoint

.. _I.10:
.. _bisect segment:

bisect segment
==============

  I.10

  To bisect a given finite straight line.

  -- Euclid


.. image:: elem.1.prop.10.png
   :align: right
   :width: 300px

Let `AB` be the given finite straight line.

Thus it is required to bisect the finite straight line `AB`.

Let the equilateral triangle `ABC` be constructed on it, [I.1] and let the angle `ACB` be bisected by the straight line `CD`; [I.9]

I say that the straight line `AB` has been bisected at the point `D`.

For, since `AC` is equal to `CB`, and `CD` is common,

- the two sides `AC`, `CD` are equal to the two sides `BC`, `CD` respectively;

and the angle `ACD` is equal to the angle `BCD`;

- therefore the base `AD` is equal to the base `BD`. [I.4]

Therefore the given finite straight line `AB` has been bisected at `D`.

- Q. E. F.

## References

[I.1]: /elem.1.3 "Book 1 - Proposition 3"
[I.4]: /elem.1.4 "Book 1 - Proposition 4"
[I.9]: /elem.1.9 "Book 1 - Proposition 9"