Construct on AD the equilateral triangle ADF.

Let ABC be the given angle.

Select the point D at random on AB.

Let AE be cut off from AC equal to AD.

 

Construct the segment AF.

We now show that riangles AFD and AFE are congruent.

 

Consider triangles AFD and AFE.

Sides AD and AE are equal. Also sides DF and EF are equal. The triangles share the common base AF. Therefore triangles AFD and AFE are congruent.

 

It follows that angles BAF and CAF are equal.

Thus the angle BAC has been bisected, which was to be proved.

QED