We claim that the triangles ACD and BCD are congruent.

For angles ACD and BCD are equal, and sides AC and BC are equal. Finally, the triangles share the common side CD.
(Prop I-4)

Let AB be the given straight line.

Construct an equilateral triangle on the base AB.
(Prop I-1)

Bisect the angle ACB.
(Prop I-9)

Extend the bisector to AB. Label the point of intersection as D.

 

The congruence of the triangles ACD and BCD implies that the sides AD and BD are equal.

Thus the line segment AB is bisected, which was to be proved.

QED