# 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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