For write-up 8, I have chosen to explore problem 13, which asks for a triangle ABC to be drawn and triangle LMN to be constructed from the intersection of the angle bisectors of ABC and the circumcenter of ABC. A diagram showing this follows:

ABC was constructed in pink, the circumcenter and angle bisectors are dotted lines, and triangle LMN is heavy lined.

The problem asked me to find the angles of LMN in terms of the angles of ABC.

The angles of LMN are the sum of half of the two angles which flank them. In this diagram, angle L is half angle A + half angle B. So on for M and N.