.. index:: construction, perpendicular

.. _I.12:
.. _construct perpendicular from point not on line:

construct perpendicular from point not on line
==============================================

  I.12

  To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line. [1]_

  -- Euclid

.. image:: elem.1.prop.12.png
   :align: right
   :width: 300px

Let `AB` be the given infinite straight line, and `C` the given point which is not on it

**PROBLEM:** draw to the given infinite straight line `AB`, from the given point `C` which is not on it, a perpendicular straight line.

For let a point `D` be taken at random on the other side of the straight line `AB`[2]_, and with centre `C` and distance `CD` let the circle `EFG` be described; [I.post.3]

- let the straight line `EG` be bisected at `H`, [I.10] and let the straight lines `CG`, `CH`, `CE` be joined. [I.post.1]

I say that `CH` has been drawn perpendicular to the given infinite straight line `AB` from the given point `C` which is not on it.

For, since `GH` is equal to `HE`, and `HC` is common,

- the two sides `GH`, `HC` are equal to the two sides `EH`, `HC` respectively;

and the base `CG` is equal to the base `CE`;

- therefore the angle `CHG` is equal to the angle `EHC`. [I.8] And they are adjacent angles.

But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. [I.def.10]

Therefore `CH` has been drawn perpendicular to the given infinite straight line `AB` from the given point `C` which is not on it.

- Q. E. F.

references
----------

[I.def.10]: /elem.1.def.10 "Book 1 - Definition 10"
[I.10]: /elem.1.10 "Book 1 - Proposition 10"
[I.8]: /elem.1.8 "Book 1 - Proposition 8"
[I.post.1]: /elem.1.post.1 "Book 1 - Postulate 1"
[I.post.3]: /elem.1.post.3 "Book 1 - Postulate 3"

footnotes
---------


.. [1] a perpendicular straight line
    , <foreign lang="greek">κάθετον εὐθεῖαν γραμμἡν</foreign>. This is the full expression for a <em>perpendicular</em>, <foreign lang="greek">κάθετος</foreign> meaning <em>let fall</em> or <em>let down</em>, so that the expression corresponds to our <em>plumb-line</em>. <foreign lang="greek">ἡ κάθετος</foreign> is however constantly used alone for a perpendicular, <foreign lang="greek">γραμμἡ</foreign> being understood.

.. [2] on the other side of the straight line AB
    , literally <quote>towards the other parts of the straight line `AB`,</quote> <foreign lang="greek">ἐπὶ τὰ ἕτερα μέρη τῆς</foreign> AB. Cf. <quote>on the same side</quote> (<foreign lang="greek">ἐπὶ τὰ αὐτὰ μέρη</foreign>) in <a href="/elem.1.post.5">Post. 5</a> and <quote>in both directions</quote> (<foreign lang="greek">ἐφ̓ ἑκάτερα τὰ μἑρη</foreign>) in <a href="/elem.1.def.23">Def. 23</a>.