The Feuerbach circle and the other remarkable circles of the cyclic polygons

It is very well known that a triangle ABC has its Feuerbach circle (nine point circle). The Feuerbach circle is a circle that passes through the midpoints EA, EB , and EC of the segments that join the vertices and the orthocenter H . These points are commonly referred to as the Euler points; we will refer the center of Feuerbach circle as F . Also, the following properties of the Feuerbach circle are very well-known; see Fig. 1: