question on circles and density

Question

If we have a circle $C(O,r)$, and a sequence of points $A_n\in C(O,r)$ for all $n>=0$ that are equidistant trigonometrically ( i.e. the lengths between two consecutive points is constant and equal to $a>0$), then how do we prove that for $k\in\mathbb N^*$ we have that among the points $A_0, A_1,...,A_k$ there exists 2 points $A_i, A_j$, $i,j\in{0,1,2,...,k}$ so that the length of the arc they form ( the least one) is less than or equal to $2\pi/k$?