What is the maximum number of sum (over all vertices) of number of distinct circles passing through at least three vertices of a convex polygon ($n$-gon), if the center of each circle required to belong to the set of vertices of the polygon?
In other words,
If we define a "centroid" by a quadruple of vertices $(a,b,c,d)$ such that $a$ is the center of a circle and three other vertices $b$, $c$, and $d$ are on circumference of this circle (so, $|ab|=|ac|=|ad|$), then what we want is the maximum number of centroids in a convex $n$-gon. (centroids are defined here https://arxiv.org/abs/1009.2218)
Any suggestion? I guess it should be of order $O(n)$. first i think there exists some 'circular order' for all the centroids around the polygon so it may be of order of n,secondly some results like Bose theorem ( see paper:"The Extremal spheres theorem " by O.Musin et al https://www.sciencedirect.com/science/article/pii/S0012365X10003997 ) suggests that number of some class of circles passing through three vertices is of order n-2. but I have no idea how to prove it.
Help me, thanks.