A conjecture about random angles formed between n points

For discussing Olympiad level Geometry Problems
User avatar
Anindya Biswas
Posts:264
Joined:Fri Oct 02, 2020 8:51 pm
Location:Magura, Bangladesh
Contact:
A conjecture about random angles formed between n points

Unread post by Anindya Biswas » Mon Jul 19, 2021 4:32 am

Let $\alpha(n)$ be the smallest non-negative angle such that for any $n\geq3$ distinct points in the plane, there exists points $A,B,C$ such that $\angle ABC\leq\alpha(n)$.
My conjecture is $\alpha(n)=\frac{\pi}{n}$ for all $n\geq3$.
If the points are in convex configuration, it's easy to prove by Pigeonhole Principle. Also $n=3,4,5,6$ cases are doable in every configuration. I could not go any further. But is this true for all $n$ and every configuration? Or is there a counter example to disprove it?
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

Post Reply