A Realisation

Geometry Level 5

Let us consider a 1126 sided polygon \( \Gamma \) (not necessarily regular) with vertices labelled \(A_{1},A_{2},...,A_{1126}\),such that the vertices \(A_{2},A_{564}\) and \(A_{1126}\), when joined,form an equilateral triangle of side length \(a\) units.

Given that the polygon \( \Gamma \) has a circumcircle, find the length of that side of \(\Gamma \) which subtends an angle of \(60^{\circ}\) at the circumcentre.

If your answer is of the form \(ka\),

input \(k\)

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