PROPOSITION 1. *PROBLEM.*

To describe an equilateral triangle on a given finite straight line.

Let *AB* be the given straight line; it is required to describe an equilateral triangle on *AB*.

From the centre *A* at the distance *AB* describe the circle *BCD*. [*Postulate* 3.

From the centre *B*, at the distance *BA*, describe the circle *ACE*. [*Postulate* 3.

From the point *C*, at which the circles cut one another, draw the straight lines *CA* and *CB* to the points *A* and *B*. [*Post*. 1.

*ABC* shall be an equilateral triangle.

Because the point *A* is the centre of the circle *BCD*, *AC* is equal to *AB*. [*Definition* 15.

And because the point *B* is the centre of the circle *ACE*, *BC* is equal to *BA*. [*Definition* 15.

But it has been shewn that *CA* is equal to *AB*;

therefore *CA* and *CB* are each of them equal to *AB*.

But things which are equal to the same thing are equal to one another. [*Axiom* 1.

Therefore *CA* is equal to *CB*.

Therefore *CA*, *AB*, BC *are equal to one another.*

Wherefore the *triangle ABC is equilateral,* [*Def*. 24.*and it is described on the given straight line AB*. q.e.f.