Construct on DE an equilateral triangle.
(Prop I-1)

Let AB be the given infinite straight line.

Let C be the given point.

Select at random another point on AB.

Label the point D.

Cut off a point E on the other side of C so that CD and CE are equal
(Prop I-3)

 

Let FD be the bisector of the angle DFE.

Then the triangles DFC and EFC are congruent. For the angles DFC and EFC are equal, the sides DF and EF are equal, and they share the common side FC.
(Prop I-4)

Therefore the angles DCF and ECF are equal.

Moreover, these angles are adjacent. Finally, because the angles are equal they must be right angles.
(Defn-10)

Therefore CF is perpendicular to AB, which was to be proved.

QED