Perfect Circles, Amicable Triangles – Extrema on Quasi-main Cevians and Main Cevians, Quasi-bisectors and Bisectors of the Angle and Poles

The family of perfect circles begins at the Fermat point, contains an incircle, and ends at the circumcircle of the reference triangle ABC. The relevant drawing presents this simple 15-cevians structure having many very interesting properties. Adding amicable triangles to this we get much more objects to research. This study deals with, among other things, the cevians mentioned in the title and the angle bisectors occurring throughout the structure. There are at least 8 characteristic points on each of three main cevians. Based on 4 of them we build cross-ratios that reach extreme values. Analyzing the entire construction we also find three pair of specific points that we can call poles. At these points intersect infinitely many circles, of which three are particularly important for us – including vertical circles with centers at vertices A, B and C.