Incenter and Circumcenter Concurrence.
File depicts a random triangle ABC. The incenter of triangle ABC was drawn and 3 circles were drawn. each circle was centered on each vertex A, B, and C, and each circle intersects at the incenter of triangle ABC. A new triangle was drawn, FGH, on the intersections of the 3 circles. The circumcenter of triangle FGH was drawn. Both the incenter of ABC and Circumcenter of FGH are on top of each other.
Angle Bisectors of triangle ABC are also Segment Bisectors of triangle FGH. The 3 segments that go through the 3 vertexes of FGH and the incenter of ABC are perpendicular to the Incenter of ABC circle.