I am looking for a proof of a problem as follows:
Let six points $A_1, A_2, A_3, A_4, A_5, A_6$ lie on a circle. Define $C(A,B)$ be any circle through points $A, B$. Let $C(A_i,A_{i+1}) \cap C(A_{i+3}, A_{i+4}) = B_i, B_{i+3}$ we take modulo 6. Let $B_1, B_2, B_3, B_4, B_5$ lie on a circle $(C)$. Let $C(A_i,A_{i+1}) \cap C(A_{i+1}, A_{i+2}) = C_{i+1}$. Show that $B_6$ also lie on the circle $(C)$ and six points $C_1, C_2, C_3, C_4, C_5, C_6$ lie on a circle.
Best Answer
Choice of coordinates
I like coordinates. The incidence relations you are interested in are invariant under Möbius transformations, so a natural setup for this problem would be the one-point compactification of the plane. Think $\mathbb{CP}^1$ if you want to, but I'll not use complex numbers for coordinates in this answer. Since your setup is invariant under Möbius transformations, you can choose specific coordinates of three points without loss of generality. So I'll choose $A_1$ to be the point at infinity, and fixed coordinates on the $x$ axis for two other points. Thus the “circle” through all the $A_i$ is the $x$ axis in my setup, and I can use parameters for the $x$ coordinates of the remaining points.
$$A_1=\infty\quad A_2=(a,0)\quad A_3=(0,0)\quad A_4=(b,0)\quad A_5=(1,0)\quad A_6=(c,0)$$
I created an interactive illustration of this specific version of your configuration, where you can move the points around and see the configuration change accordingly. I alternated between fixed and variable points in an attempt to make things more symmetric, in the hope of making individual expressions not too complicated.
Next we can pick an arbitrary point $B_1=(d,e)$ anywhere in the plane. This results in the
$$B_4=\frac1{(a-d)^2+e^2}\begin{pmatrix} (a(b-d)+(a-b))(a-d)+ae^2 \\ (a-b)(a-1)e \end{pmatrix}$$
computed as the second point of intersection between the corresponding circles.
First condition for cocircularity of the Bi
Choosing $B_2=(x,y)$ will define the circle through $B_1,B_2,B_4$. On the other hand this allows us to construct $B_5$ as the intersection of $C(A_2,A_3)$ and $C(A_5,A_6)$. We can and check whether that point lies on the same circle. In general it does not. The condition for $B_1,B_2,B_4,B_5$ to be cocircular can be factored into the following independent conditions:
$$ a=1 \\\vee\\ (d-a)^2+e^2=(1-a)(b-a) \\\vee\\ (a-c-1)(x^2+y^2) + 2cx = ac \\\vee\\ (a-c)e(x^2+y^2) + (-ab + bc - a + b)ex\\ + ((a-c)((b-d)d - e^2) - (a-b)(c-d))y + (a-b)ce=0 $$
The first three of these conditions can be seen as non-degeneracy constraints. If $a=1$ then $A_2=A_5$, which is a degenerate situation. The second condition characterizes the situations where $B_1$ and $B_4$ would coincide, while the third is when $B_2$ and $B_5$ would be the same. So if we can assume that all points in the configuration are distinct, then we can concentrate on the last condition.
It describes a circle, so we now know that the point $B_2$ must lie on a certain circle if $B_5$ is to be cocircular with $B_1,B_2,B_4$. Actually this circle already passes through $B_1$ and $B_4$. So knowing all the $A_i$ and $B_1$ already fixes the circle on which all the $B_i$ must lie.
Second condition for cocircularity of the Bi
We can play the same game for $B_3$ and $B_6$ instead of $B_2$ and $B_5$. What's the conditions for $B_1,B_3,B_4,B_6$ being cocircular if we choose $B_3=(x',y')$?
$$ (d-a)^2+e^2=(1-a)(b-a) \\\vee\\ (x'-c)^2+y'^2=(c-b)c \\\vee\\ (a-c)e(x'^2+y'^2) + (-ab + bc - a + b)ex'\\ + ((a-c)((b-d)d - e^2) - (a-b)(c-d))y' + (a-b)ce=0 $$
The last conditon is the same as above, so if the cocircularity condition is satisfied for $B_1,B_2,B_4,B_5$ and all points are distinct, and if $B_3$ is chosen from that same circle, too, then the cocircularity condition must be satisfied for $B_1,B_3,B_4,B_6$ as well, and all six $B_i$ are cocircular.
Conditions for cocircularity of the Ci
A similar game can be played for the $C_i$ as well. Their coordinates are considerably more complicated, though, and so are the conditions for cocircularity of four of these. But assuming the $B_i$ to be cocircular (i.e. assuming the respective last condition to hold for $B_2$ and $B_3$), one can demonstrate that all the points must be cocircular, too. In particular, if the condition for $B_2$ holds, then one can show that $(C_1,C_2,C_5,C_6)$ and $(C_2,C_3,C_4,C_5)$ are cocircular, and if the condition for $B_3$ holds, then $(C_1,C_2,C_3,C_4)$ and $(C_1,C_4,C_5,C_6)$ are cocircular. If both hold, then all points must be cocircular. You could start by defining the circle via $C_1,C_2,C_3$, then add $C_4$ due to $(C_1,C_2,C_3,C_4)$, add $C_5$ due to $(C_2,C_3,C_4,C_5)$ and add $C_6$ due to $(C_1,C_4,C_5,C_6)$. All again assuming distinct points, of course.
Full log of computations
I computed the things I described above with a lot of help from Sage. I included the full log of my Sage session, including descriptions, in the blog post I created for the interactive illustration.