.. index:: construction, lines

.. _I.2:
.. _construct equal segments by extension:

construct equal segments by extension
=====================================

  I.2

  To place at a given point (as an extremity) a straight line equal to a given
  straight line.

  -- Euclid

.. image:: construct-equal-segments-by-extension.png
   :align: right
   :width: 300px


From the given points :math:`A` and :math:`B` :ref:`set segment`
:math:`\bar{AB}`.

From the given points :math:`C` and :math:`D` :ref:`set segment`
:math:`\bar{CD}` shorter than :math:`\bar{AB}`.

**PROBLEM:** Extend segment :math:`\bar{CD}` to a length equal to :math:`\bar{AB}`

:ref:`construct circle` :math:`(A, B)` as :math:`c_1`

.. :ref:`construct circle` :math:`(C, D)` as :math:`c_2`

With points :math:`A` and :math:`C` :ref:`construct segment` :math:`\bar{AC}`

:ref:`contruct equilateral triangle` on :math:`\bar{AC}`

identify new point of equaliteral triangle  as :math:`E`

:ref:`construct line` :math:`\overline{EA}`

:ref:`construct line` :math:`\overline{EC}`

Identify new point on :math:`c_1` intersecting with :math:`\overline{EA}`  as :math:`F`

:ref:`construct circle` :math:`(E, F)`` as :math:`c_2`

Identify new point on :math:`c_2` intersecting with :math:`\overline{EC}`  as :math:`G`



Let the straight lines **AE**, **BF** be produced [2]_ in a straight line
with **DA**, **DB**; [I.post.2] with centre **B** and distance **BC** let the
circle **CGH** be described; [I.post.3] and again, with centre **D** and
distance **DG** let the circle **GKL** be described. [I.post.3]

Then, since the point **B** is the centre of the circle **CGH**, 

:math:`\therefore` **BC** is equal to **BG**.

Again, since the point **D** is the centre of the circle **GKL**, 

:math:`\therefore` **DL** is equal to **DG**.

And in these **DA** is equal to **DB**; 

:math:`\therefore` the remainder **AL** is equal to the remainder **BG**  [3]_.
  [I.c.n.3]

But **BC** was also proved equal to **BG**; 

:math:`\therefore` each of the straight lines **AL**, **BC** is equal to **BG**.

And things which are equal to the same thing are also equal to one another;
[I.c.n.1] 

:math:`\therefore` **AL** is also equal to **BC**.

:math:`\therefore` at the given point **A** the straight line **AL** is placed equal to
the given straight line **BC**.

- (Being) what it was required to do.

dependencies
------------

[I.def.15]: /elem.1.def.15 "Book I - Definition 15"
[I.1]: /elem.1.1 "Book I - Proposition 1"
[I.post.1]: /elem.1.post.1 "Book I - Postulate 1"
[I.post.2]: /elem.1.post.2 "Book I - Postulate 2"
[I.post.3]: /elem.1.post.3 "Book I - Postulate 3"
[I.c.n.1]: /elem.1.c.n.1 "Book I - Common Notion 1"
[I.c.n.3]: /elem.1.c.n.3 "Book I - Common Notion 3"




footnotes
---------


.. [1] (as an extremity).  

   I have inserted these words because <quote>to place a straight line
   <em>at</em> a given point</quote> (<foreign lang="greek">πρὸς τῷ δοθέντι
   σημείῳ</foreign>) is not quite clear enough, at least in English.

.. [2] Let the straight lines AE, BF be produced....  

   It will be observed that in this first application of <a
   href="/elem.1.post.2">Postulate 2</a>, and again in <a href="/elem.1.5">I.
   5</a>, Euclid speaks of the <em>continuation</em> of the straight line as
   that which is produced in such cases, <foreign
   lang="greek">ἐκβεβλήσθωσαν</foreign> and <foreign
   lang="greek">προσεκβεβλήσθωσαν</foreign> meaning little more than
   <em>drawing</em> straight lines <quote>in a straight line with</quote> the
   given straight lines. The first place in which Euclid uses phraseology
   exactly corresponding to ours when <pb n="245"/>speaking of a straight line
   being produced is in <a href="/elem.1.16">I. 16</a>: <quote>let one side of
   it, **BC**, be produced to **D** </quote> (<foreign
   lang="greek">προσεκβεβλήσθω αὐτοῦ μία πλευρὰ ἡ ΒΓ ἐπὶ τὸ Δ</foreign>).

.. [3] the remainder AL...the remainder BG.  
   The Greek expressions are <foreign lang="greek">λοιπὴ ἡ ΑΛ</foreign> and
   <foreign lang="greek">λοιπῇ τῇ</foreign> BH, and the literal translation
   would be <quote>**AL** (or **BG**) <em>remaining</em>,</quote> but the shade
   of meaning conveyed by the position of the definite article can hardly be
   expressed in English.