This proposition constructs a point bisecting a given line. While the previous proposition leverages SSS congruence, this one leverages SAS congruence. The equilateral triangle is one of the most powerful tools for constructing congruences because of its symmetries. Even though half of these are lost in hyperbolic geometry, it's still instinctive to construct an equilateral triangle off AB, and call the resulting point C. This sets up the construction of an angle bisector of angle C, and the intersection between this and AB, D, would then bisect AB, because △ACD is congruent to △BCD.