Euclid I-9. (pg 8)

Emily Crum
Explanation: Let angle BAC be the rectilinear angle. Create point D on BC. Then make pt. E on CA such that AD = AE. Join pts. D and E together with a line segment. Next create an equilateral triangle with DE. (This was done previously and does not need to be proven) This equilateral triangle is triangle FDE Connect F and A. Triangles FDA and FEA are congruent because of SSS AD = AE as per the earlier directions AF = AF (AF is in common between the two) and DF = EF because Triangle FDE is equilateral. Because these two triangles are congruent, this means that all of their angles are congruent too Thus, angle BAF = angle CAF. Since the two angles equal each other, then Angle BAC has been bisected.