The lively siblings of the Pentagon theorem

The five circles in the classical Pentagon theorem of Miquel are given as circumcircles of five certain triangles in the pentagon. If one chooses instead the circumcircles of five other triangles, one gets a different configuration of circles. This resulting configuration of circles carries three families of five concyclic quadruples of points. Together with the five circumcircles this gives a total of 20 circles. The radical axes of each two of these twenty circles are all concurrent.


Introduction
The classical version of Miquel's Pentagon theorem on the Riemann sphere can be formulated as follows: The situation is shown in Fig. 1. Miquel's original proof can be found in [4,Théorème III]. It is based on classical angle theorems. A computer assisted algebraic proof which uses null bracket algebra has been published in [3]. A simple algebraic proof based on the cross ratio has been discussed in [1].
The assumption that the Möbius circles h i intersect (not touch) each other in I implies that the points q i and s i are different from I. In addition, since we This research received no specific grant from any funding agency.  Figure 1 The classical Pentagon theorem assume that any three of the circles h i only meet in I, we have that the 10 points q i , s i are pairwise distinct. These assumptions can be relaxed if one is interested in degenerate cases of the configuration. Using the Möbius transformation z → 1/(z − I) we may always assume that I = ∞. In this case, the Möbius circles h i are lines in the complex plane.
The idea is now to replace the circles k i through s i , q i−1 , q i+1 by circles k i through s i , q i−2 , q i+2 . This variant has apparently not yet been treated in the literature. Surprisingly, this configuration shows numerous incidences, even significantly more than the classical Pentagon Theorem 1.  Fig. 3). Fig. 4).
It does not matter which of the two points of intersection of the circles k i and k j is denoted by p ij or r ij . However, let us agree that the intersection of k i−2 and k i+2 is q i = r i+2,i−2 while the second point of intersection of k i−2 and k i+2 will be denoted p i+2,i−2 . Note that k i−2 and k i+2 always have two points of intersection, q i = r i+2,i−2 and p i+2,i−2 . However, whether k i and k i+2 have common points depends on the position of the points q i . Interestingly, not only the Miquel Pentagon Theorem has this new relative in Theorem 2, but recently also a new variant of Morley's Five Circles Theorem was discovered (see [2]).
It turns out that Theorem 2 follows easily from another incidence result, the mother of the siblings in Fig. 5, which we formulate now.

Siblings of the Pentagon Theorem: The proof
In this section, we carry out the computations in the complex plane C. In particular, z denotes a complex variable, andz is its complex conjugate. The equation of a line through two different points p, q is given by Similarly, the equation of a circle through three different points p, q, r (which do not lie on a line) is given by since z = p, z = q, z = r are solutions of this equation, and expanded it has the form Observe that the group of Möbius transformations The point s i = ∞ is the intersection of the lines h i−2 and h i+2 . Solving the corresponding linear system of the two equations yields The equation of the block k i through the points s i , q i−2 , q i+2 is then given by p 24 J Figure 5 The mother of the siblings By expanding the products we can bring this equation into the standard form with Using (1) in these formulas we obtain The equation of the radical axis a ij of the circles k i and k j results by eliminating zz from their respective equations (2). We obtain