construct a parallelogram on a segment equal to a given triangle
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.. index:: construction, parallelogram

.. image:: elem.1.prop.44.png
   :align: right
   :width: 300px

To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle.

===

Let `AB` be the given straight line, `C` the given triangle and `D` the given rectilineal angle; thus it is required to apply to the given straight line `AB`, in an angle equal to the angle `D`, a parallelogram equal to the given triangle `C`.

Let the parallelogram `BEFG` be constructed equal to the triangle `C`, in the angle `EBG` which is equal to `D` [I.42]; let it be placed so that `BE` is in a straight line with `AB`; let<pb n="342"/>`FG` be drawn through to `H`, and let `AH` be drawn through `A` parallel to either `BG` or `EF`. [I.31]

Let `HB` be joined. 

Then, since the straight line `HF` falls upon the parallels `AH`, `EF`, [^I.44:1]

- the angles `AHF`, `HFE` are equal to two right angles. [I.29]

Therefore the angles `BHG`, `GFE` are less than two right angles; and straight lines produced indefinitely from angles less than two right angles meet; [I.post.5] 

- therefore `HB`, `FE`, when produced, will meet.

Let them be produced and meet at `K`; through the point `K` let `KL` be drawn parallel to either `EA` or `FH`, [I.31] and let `HA`, `GB` be produced to the points `L`, `M`. 

Then `HLKF` is a parallelogram, `HK` is its diameter, and `AG`, `ME` are parallelograms. and `LB`, `BF` the so-called complements, about `HK`; 

- therefore `LB` is equal to `BF`. [I.43]

But `BF` is equal to the triangle `C`; 

- therefore `LB` is also equal to `C`. [I.c.n.1]

And, since the angle `GBE` is equal to the angle `ABM`, [I.15] 

- while the angle `GBE` is equal to `D`, the angle `ABM` is also equal to the angle `D`.

Therefore the parallelogram `LB` equal to the given triangle `C` has been applied to the given straight line `AB`, in the angle `ABM` which is equal to `D`.

- Q. E. F.

## References

[I.15]: /elem.1.15 "Book 1 - Proposition 15"
[I.29]: /elem.1.29 "Book 1 - Proposition 29"
[I.31]: /elem.1.31 "Book 1 - Proposition 31"
[I.42]: /elem.1.42 "Book 1 - Proposition 42"
[I.43]: /elem.1.43 "Book 1 - Proposition 43"
[I.post.5]: /elem.1.post.5 "Book 1 - Postulate 5"
[I.c.n.1]: /elem.1.c.n.1 "Book 1 - Common Notion 1"

## Footnotes

[^I.44:1]:since the straight line HF falls...."
    The verb is in the aorist (<foreign lang="greek">ὲνέπεσεν</foreign>) here and in similar expressions in the following propositions.