Circle Cutting an Equilateral Triangle

Geometry Level 5

A circle \(\Gamma\) cuts the sides of a equilateral triangle \(ABC \) at \(6\) distinct points. Specifically, \(\Gamma\) intersects \(AB\) at points \(D\) and \(E\) such that \(A, D, E, B\) lie in order. \(\Gamma\) intersects \(BC\) at points \(F\) and \(G\) such that \(B, F, G, C\) lie in order. \(\Gamma\) intersects \(CA\) at points \(H\) and \(I\) such that \(C, H, I, A\) lie in order. If \( |AD| =3\), \(|DE| =39\), \(|EB| = 6\) and \( |FG| = 21 \), what is the value of \(|HI|^2 \)?

Details and assumptions

\(|\cdot|\) notation denotes the straight line distance between points and not the arc length distance.

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