.. index:: construction, triangles

.. _I.22:
.. _construct triangle from three segments:

construct a triangle from three segments
========================================

  I.22

  Out of three straight lines, which are equal to three given straight lines,
  to construct a triangle: thus it is necessary [2]_ that two of the
  straight lines taken together in any manner should be greater than the
  remaining one. [I.20] [1]_ [3]_

  -- Euclid

.. image:: elem.1.prop.22.png
   :align: right
   :width: 300px

Let the three given straight lines be `A`, `B`, `C`, and of these let two taken together in any manner be greater than the remaining one, namely `A`, `B` greater than `C`,

- `A`, `C` greater than `B`,

and `B`, `C` greater than `A`; thus it is required to construct a triangle out of straight lines equal to `A`, `B`, `C`.


Let there be set out a straight line `DE`, terminated at `D` but of infinite length in the direction of `E`, and let `DF` be made equal to `A`, `FG` equal to `B`, and `GH` equal to `C`. [I.3]

With centre `F` and distance `FD` let the circle `DKL` be described; again, with centre `G` and distance `GH` let the circle `KLH` be described; and let `KF`, `KG` be joined;

I say that the triangle `KFG` has been constructed out of three straight lines equal to `A`, `B`, `C`.

For, since the point `F` is the centre of the circle `DKL`,

- `FD` is equal to `FK`.

But `FD` is equal to `A`;

- therefore `KF` is also equal to `A`.

Again, since the point `G` is the centre of the circle `LKH`,

- `GH` is equal to `GK`.

But `GH` is equal to `C`;

- therefore `KG` is also equal to `C`. And `FG` is also equal to `B`;

therefore the three straight lines `KF`, `FG`, `GK` are equal to the three straight lines `A`, `B`, `C`.

Therefore out of the three straight lines `KF`, `FG`, `GK`, which are equal to the three given straight lines `A`, `B`, `C`, the triangle `KFG` has been constructed.

- Q. E. F.

references
----------

[I.3]: /elem.1.3 "Book 1 - Proposition 3"
[I.20]: /elem.1.20 "Book 1 - Proposition 20"

footnotes
---------

.. [1] enunciation note

   This is the first case in the <title>Elements</title> of a <foreign
   lang="greek">διορισμός</foreign> to a problem in the sense of a statement of
   the conditions or limits of the possibility of a solution. The criterion is
   of course supplied by the preceding proposition.

.. [2] thus it is necessary.

   This is usually translated (e.g. by Williamson and Simson)
   <quote><title>But</title> it is necessary,</quote> which is however
   inaccurate, since the Greek is not <foreign lang="greek">δεῖ δέ</foreign>
   but <foreign lang="greek">δεῖ δή</foreign>. The words are the same as those
   used to introduce the <foreign lang="greek">διορισμός</foreign> in the other
   sense of the <quote>definition</quote> or <quote>particular
   statement</quote> of a construction to be effected. Hence, as in the latter
   case we say <quote>thus it is required</quote> (e.g. to bisect the finite
   straight line `AB`, <a href="/elem.1.10">I. 10</a>), we should here
   translate <quote><em>thus</em> it is necessary.</quote>

.. [3] enunciation note
   
   To this enunciation all the MSS. and Boethius add, after the <foreign
   lang="greek">διορισμός</foreign>, the words <quote>because in any triangle
   two sides taken together in any manner are greater than the remaining
   one.</quote> 
   
   But this explanation has the appearance of a gloss, and it is
   omitted by Proclus and Campanus. Moreover there is no corresponding addition
   to the <foreign lang="greek">διορισμός</foreign> of <a href="/elem.6.28">VI.
   28</a>.