.. index:: proof, triangles

.. _I.20:
.. _any two sides of a triangle are greater than the remaining side:

any two sides of a triangle are greater than the remaining side
===============================================================

  I.20

  In any triangle two sides taken together in any manner are greater than the remaining one. [1]_

  -- Euclid


.. image:: elem.1.prop.20.png
   :align: right
   :width: 300px

For let `ABC` be a triangle; I say that in the triangle `ABC` two sides taken
together in any manner are greater than the remaining one, namely

- `BA`, `AC` greater than `BC`, `AB`, `BC` greater than `AC`, `BC`, `CA` greater than `AB`.

For let `BA` be drawn through to the point `D`, let `DA` be made equal to `CA`, and let `DC` be joined.

Then, since `DA` is equal to `AC`, the angle `ADC` is also equal to the angle `ACD`; [I.5]

- therefore the angle `BCD` is greater than the angle `ADC`. [I.c.n.5]

And, since `DCB` is a triangle having the angle `BCD` greater than the angle `BDC`,

- and the greater angle is subtended by the greater side, [I.19] therefore `DB` is greater than `BC`.

But `DA` is equal to `AC`;

- therefore `BA`, `AC` are greater than `BC`.

Similarly we can prove that `AB`, `BC` are also greater than `CA`, and `BC`, `CA` than `AB`.

Therefore etc.

- Q. E. D.


references
----------

[I.5]: /elem.1.5 "Book 1 - Proposition 5"
[I.19]: /elem.1.19 "Book 1 - Proposition 19"
[I.c.n.5]: /elem.1.c.n.5 "Book 1 - Common Notion 5"


footnotes
---------

.. [1] note

   It was the habit of the Epicureans, says Proclus (<xref n="Proc. p. 322"
   from="ROOT" to="DITTO">p. 322</xref>), to ridicule this theorem as being
   evident even to an ass and requiring no proof, and their allegation that the
   theorem was <quote>known</quote> (<foreign lang="greek">γνώριμον</foreign>)
   even to an ass was based on the fact that, if fodder is placed at one
   angular point and the ass at another, he does not, in order to get to his
   food, traverse the two sides of the triangle but only the one side
   separating them (an argument which makes Savile exclaim that its authors
   were <quote><foreign lang="la">digni ipsi, qui cum Asino foenum
   essent</foreign>,</quote> p. 78). Proclus replies truly that a mere
   perception of the truth of the theorem is a different thing from a
   scientific proof of it and a knowledge of the reason <em>why</em> it is
   true. Moreover, as Simson says, the number of axioms should not be increased
   without necessity.