Circle Cutting an Equilateral Triangle

Geometry Level 5

A circle Γ\Gamma cuts the sides of a equilateral triangle ABCABC at 66 distinct points. Specifically, Γ\Gamma intersects ABAB at points DD and EE such that A,D,E,BA, D, E, B lie in order. Γ\Gamma intersects BCBC at points FF and GG such that B,F,G,CB, F, G, C lie in order. Γ\Gamma intersects CACA at points HH and II such that C,H,I,AC, H, I, A lie in order. If AD=3 |AD| =3, DE=39|DE| =39, EB=6|EB| = 6 and FG=21 |FG| = 21 , what is the value of HI2|HI|^2 ?

Details and assumptions

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

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