.. index:: triangle

.. _!.21:

Proposition I.21
================


  I.21

  If on one of the sides of a triangle, from its extremities, there be
  constructed two straight lines meeting within the triangle, [1]_ the
  straight lines so constructed [2]_ will be less than the remaining two
  sides of the triangle, but will contain a greater angle.

  -- Euclid

.. todo:: update title

.. image:: elem.1.prop.21.png
   :align: right
   :width: 300px


On `BC`, one of the sides of the triangle `ABC`, from its extremities `B`, `C`, let the two straight lines `BD`, `DC` be constructed  meeting within the triangle;

I say that `BD`, `DC` are less than the remaining two sides of the triangle `BA`, `AC`, but contain an angle `BDC` greater than the angle `BAC`.

For let `BD` be drawn through to `E`.

Then, since in any triangle two sides are greater than the remaining one, [I.20]

- therefore, in the triangle `ABE`, the two sides `AB`, `AE` are greater than `BE`.

Let `EC` be added to each;

- therefore `BA`, `AC` are greater than `BE`, `EC`.

Again, since, in the triangle `CED`,

- the two sides `CE`, `ED` are greater than `CD`, let `DB` be added to each; therefore `CE`, `EB` are greater than `CD`, `DB`.

But `BA`, `AC` were proved greater than `BE`, `EC`;

- therefore `BA`, `AC` are much greater than `BD`, `DC`.

Again, since in any triangle the exterior angle is greater than the interior and opposite angle, [I.16] therefore, in the triangle `CDE`,

- the exterior angle `BDC` is greater than the angle `CED`.

For the same reason, moreover, in the triangle `ABE` also,

- the exterior angle `CEB` is greater than the angle `BAC`.

But the angle `BDC` was proved greater than the angle `CEB`;

- therefore the angle `BDC` is much greater than the angle `BAC`.

Therefore etc.

- Q. E. D.

references
----------

[I.20]: /elem.1.20 "Book 1 - Proposition 20"
[I.16]: /elem.1.16 "Book 1 - Proposition 16"

footnotes
---------

.. [1] be constructed...meeting within the triangle.

   The word <quote>meeting</quote> is not in the Greek, where the words are
   <foreign lang="greek">ἐντὸς συσταθῶσιν. συνίστασθαι</foreign> is the word
   used of constructing two straight lines <em>to a point</em> (cf. <a
   href="/elem.1.7">I. 7</a>) or so as to form a triangle; but it is necessary
   in English to indicate that they <em>meet</em>.

.. [2] the straight lines so constructed.
   
   Observe the elegant brevity of the Greek <foreign lang="greek">αἱ συσταθεῖσαι</foreign>.