.. index:: proof, triangles

.. _I.24:
.. _similar triangles 2:

similar triangles
=================

  I.24

  If two triangles have the two sides equal to two sides respectively, but have
  the one of the angles contained by the equal straight lines greater than the
  other, they will also have the base greater than the base. 

  -- Euclid

.. image:: elem.1.prop.24.png
   :align: right
   :width: 300px

Let `ABC`, `DEF` be two triangles having the two sides `AB`, `AC` equal to the
two sides `DE`, `DF` respectively, namely `AB` to `DE`, and `AC` to `DF`, and
let the angle at `A` be greater than the angle at `D`;

I say that the base `BC` is also greater than the base `EF`. [1]_

For, since the angle `BAC` is greater than the angle `EDF`, let there be
constructed, on the straight line `DE`, and at the point `D` on it, the angle
`EDG` equal to the angle `BAC`; [I.23] let `DG` be made equal to either of the
two straight lines `AC`, `DF`, and let `EG`, `FG` be joined. 

Then, since `AB` is equal to `DE`, and `AC` to `DG`, the two sides `BA`, `AC`
are equal to the two sides `ED`, `DG`, respectively; 

- and the angle `BAC` is equal to the angle `EDG`; therefore the base `BC` is
  equal to the base `EG`. [I.4]

Again, since `DF` is equal to `DG`, 

- the angle `DGF` is also equal to the angle `DFG`; [I.5] therefore the angle
  `DFG` is greater than the angle `EGF`.

Therefore the angle `EFG` is much greater than the angle `EGF`.

And, since `EFG` is a triangle having the angle `EFG` greater than the angle
`EGF`, 

- and the greater angle is subtended by the greater side, [I.19] the side `EG`
  is also greater than `EF`.

But `EG` is equal to `BC`. 

- Therefore `BC` is also greater than `EF`.

Therefore etc.

- Q. E. D.

references
----------

[I.4]: /elem.1.4 "Book 1 - Proposition 4"
[I.5]: /elem.1.5 "Book 1 - Proposition 5"
[I.19]: /elem.1.19 "Book 1 - Proposition 19"
[I.23]: /elem.1.23 "Book 1 - Proposition 23"

footnotes
---------

.. [1] I say that the base `BC` is also greater than the base `EF`. 

    I have naturally left out the well-known words added by Simson in order to
    avoid the necessity of considering three cases: <quote>Of the two sides
    `DE`, `DF` let `DE` be the side which is not greater than the
    other.</quote>

    I doubt whether Euclid could have been induced to insert the words himself,
    even if it had been represented to him that their omission meant leaving
    two possible cases out of consideration. His habit and that of the great
    Greek geometers was, not to set out all possible cases, but to give as a
    rule one case, generally the most difficult, as here, and to leave the
    others to the reader to work out for himself. We have already seen one
    instance in <a href="/elem.1.7">I. 7</a>.