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Assume you have a regular polygon ( $n$-sides). and Let $A=\{ x_0, x_2, \cdots , x_{n-1} \}$ be vertices of the polygon.

My Question is:

Are there is any formula that tell us what is the angle between any three distincts vertices let say $x_i, x_j, x_k$?

As a special case we know the angle made by $x_m,x_{m+1},x_{m+2}$ in $n$-sided polygon is $\frac{180(n-2)}{n}$.

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  • $\begingroup$ Do you mean a regular polygon? $\endgroup$ – Andrei May 8 at 5:10
  • $\begingroup$ yes. I just fixed it. $\endgroup$ – henry May 8 at 5:11
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Let's assume that $i<j<k$. Also note that all points in $A$ are on a circle, and the angle from $x_a$, center of the circle $O$, and $x_{a+1}$ is $\frac{360^\circ}{n}$. Then the angle between $x_j$, $O$, $x_i$ is $\frac{360^\circ}{n}(j-i)$. This is an isosceles triangle, so the angle $\angle Ox_jx_i$ is $\frac 12(180^\circ-\frac{360^\circ}{n}(j-i))$. Similarly $\angle x_kx_jO=\frac 12(180^\circ-\frac{360^\circ}{n}(k-j))$. Then just add them together to get$$\angle x_kx_jx_i=180^\circ-\frac{180^\circ}{n}(k-i)$$ Note that the angle depends only on the end points.

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