Point $M$ is located inside a regular $n$-gon. Prove that there exist different vertices $A$ and $B$ that $$180^{\circ}\left(1-\frac1n\right)\le \angle{AMB}\le 180^{\circ}$$
My work so far:
Let $d(M,A_iA_j) -$ the distance from a point $M$ to $A_iA_j$. Let $A,B$ such that $d(M,AB)=\min d(M,A_iA_j)$.
I want to prove that $180^{\circ}\left(1-\frac1n\right)\le \angle{AMB}$. I need help here.
Consider a regular $n-$gon inscribed in a circle with centre $O$. $A$ and $B$ are two of it's vertices and $M$ is a point located on/in any one of the $n$ equilateral triangles
It can be seen by exterior angle theorem, that for this regular $n-$ gon, $\frac{360}{n} \leq \angle{AMB} \leq 180^{\circ}.$
But we are required to prove that $180 \cdot (1-\frac{1}{n}) \leq \angle{AMB} \leq 180^{\circ}.$
So, by placing the point $M$ inside one of the $n$ equilateral triangles, we just need to to prove that $\frac{360}{n} \leq 180\cdot (1-\frac{1}{n})\Rightarrow n \geq 3$, which is indeed true for all regular polygons.