The following is as given in Sir Thomas L. Heath's translation, which can be found in the book The Thirteen Books of The Elements, Vol. 1.

Book I | Book II | Book IX |
---|---|---|

Definitions, Postulates, and Common Notions | Definitions | Proposition 20 |

Proposition 1, Proposition 3, | Proposition 14 | Proposition 36 |

Proposition 5, Proposition 6, | ||

Proposition 29, Proposition 47 |

*On a given finite straight line to construct and equilateral triangle.*

Let `AB` be the given finite straight line.

Thus it is required to construct an equilateral triangle on the straight line `AB`.

With centre `A` and distance `AB` let the circle `BCD` be described; [Post. 3]

again, with centre `B` and distance `BA` let the circle `ACE` be described; [Post. 3]

and from the point `C`, in which the circles cut one another, to the points `A`, `B` let the straight lines `CA`, `CB` be joined.
[Post. 1]

Now, since the point `A` is the centre of the circle `CDB`,

Again, since the point `B` is the centre of the circle `CAE`,

But `CA` was also proved equal to `AB`;

therefore each of the straight lines `CA`, `CB` is equal to `B`.

And things which are equal to the same thing are also equal to one another; [C. N. 1]

therefore `CA` is also equal to `CB`.

Therefore the three straight lines `CA`, `AB`, `BC` are equal to one another.

Therefore the triangle `ABC` is equilateral; and it has been constructed on the given finite straight line `AB`.

(Being) what it was required to do.