Consider $n$ points evenly spaced around a circle. Connect pairs of points that are the furthest away from each other with $k$ arcs connecting each pair of points (these arcs could be pictured as evenly spaced geodesics projected straight down from the surface of a hemisphere onto the unit disk). In the interior of the disk what is maximum number of lines that intersect at one point (not including the center of the circle)? I know that the degree of intersection at the center of the circle is equal to $n/2.$
I tried drawing diagrams and thinking about this a lot but could not come to an answer. I think maybe the most lines that meet at a common point in the interior of the disk excluding the center of the circle is two or three but this is just a guess. Is there an easy way to get the correct answer?
Here's a diagram I made to visualise what's going on:
