Cyclologic Triangles

Two nondegenerate triangles and are cyclologic if the three circles through , , and are concurrent. In this case, the circles through , , and are also concurrent, and conversely. The two points of concurrence are called the cyclologic centers of the triangles (Rabinowitz and Suppa 2026, p. 105).

Let be any point inside . For equal to 13, 14, 15, or 16, let , , and be the Kimberling centers of , , and , respectively. Then and are cyclologic (Rabinowitz and Suppa 2026). The four choices are the first Fermat point , second Fermat point , first isodynamic point , and second isodynamic point , respectively.

For , an inversion centered at maps the three configurations to equilateral triangles erected outwardly on the sides of the inverse of . Their circumcircles are concurrent at the first Fermat point by Napoleon's theorem. Applying the inversion again gives three concurrent circles through , , and . The assumption that is interior is essential to the outward-orientation step: each first isodynamic point lies inside the corresponding circumcircle, while the opposite vertex lies outside it. Consequently, the result for can fail when lies outside . For , the analogous inversion gives equilateral triangles erected inwardly, and their circumcircles concur at the second Fermat point.