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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