Proposition 9

This proposition constructs a ray bisecting a given angle. So far, most of the constructions have only involved sides, you can't make use of the previous constructions directly. However, Euclid did prove a powerful tool for converting equivalences in side lengths to equivalences in angles: SSS congruence. To construct the two congruent triangles, Euclid selects an arbitrary point D on AB, and transfers it onto AC. By construction, this creates one side congruence, and another of the side congruences will be the bisector itself. Equilateral triangles have three sides congruent, so the most natural choice is to construct an equilateral triangle off ED, and to call the resulting point F. AF then forms the angle bisector.