What is the largest possible value of the smallest angle between the diagonals of an $n$-gon for even $n$?

Question

In this answer, I showed that the maximum value of the smallest angle between two diagonals of a $21-$gon is $8\frac{4}{7}$ degrees. The method given generalizes to all polygons with an odd number of sides, other than $3,$ giving an angle of $180/n$ degrees. I also think that $180/n$ is optimal for even numbers other than $4,$ since a regular polygon achieves this angle. For $4,$ the optimal angle is $90$ degrees, which is achieved by a square. What is the largest possible value of the smallest angle between the diagonals of an $n$-gon for even $n$?

Answer 1

We can look at the diagonals forming one or two stars which connect two vertices $n-1$ edges away if the number of vertices of the polygon is $2n$ with $n > 2.$ Apart from pairs of such diagonals, every diagonal intersects the other diagonals either at a vertex or inside the polygon. Therefore, these account for at least $n$ directions of diagonals. To find $n$ more, we can look at the diagonals connecting two vertices $n$ edges away. Since all these diagonals intersect each other inside the polygon and the previous diagonals either at a vertex or inside the polygon, there are $n$ new directions formed. Therefore, there are at least $2n$ directions of diagonals, so the minimum angle between diagonals is at most $180/2n.$