Elements I.1

Euclid's Elements: Book I: Proposition 1

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, AC is equal to AB. Def. 15
Again, since the point B is the centre of the circle CAE, BC is equal to BA. Def. 15

But CA was also proved equal to AB;
therefore each of the straight lines CA, CB is equal to AB.

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.