.. index:: construction, angle, bisect

.. _I.9:
.. _bisect angle:

bisect angle
============

  I.9

  To bisect a given rectilineal angle.

  -- Euclid


.. image:: elem.1.prop.9.png
   :align: right
   :width: 300px

Let the angle `BAC` be the given rectilineal angle.

Thus it is required to bisect it.

Let a point `D` be taken at random on `AB`; let `AE` be cut off from `AC` equal to `AD`; [I.3] let `DE` be joined, and on `DE` let the equilateral triangle `DEF` be constructed; let `AF` be joined.

I say that the angle `BAC` has been bisected by the straight line `AF`.

For, since `AD` is equal to `AE`, and `AF` is common,

- the two sides `DA`, `AF` are equal to the two sides `EA`, `AF` respectively.

And the base `DF` is equal to the base `EF`;

- therefore the angle `DAF` is equal to the angle `EAF`. [I.8]

Therefore the given rectilineal angle `BAC` has been bisected by the straight line `AF`.

- Q. E. F.

references
----------

[I.3]: /elem.1.3 "Book 1 - Proposition 3"
[I.8]: /elem.1.8 "Book 1 - Proposition 8"