Let $A_1A_2\ldots A_n$ be a regular hexagon and $M$ be a point on the shorter arc $A_1A_n$ of its circumcircle. Prove that the value of $$\frac{A_2M+A_3M+\ldots+A_{n-1}M}{A_1M+A_nM}$$is constant and find this value.
Source: Moldova 2000 Grade 10 P4
Tags: ratio, geometry
Let $A_1A_2\ldots A_n$ be a regular hexagon and $M$ be a point on the shorter arc $A_1A_n$ of its circumcircle. Prove that the value of $$\frac{A_2M+A_3M+\ldots+A_{n-1}M}{A_1M+A_nM}$$is constant and find this value.