Question 135
A regular octahedron $A B C D E F$ is given such that $A D$, $B E$, and $C F$ are perpendicular. Let $G$, $H$, and $I$ lie on edges $A B$, $B C$, and $C A$ respectively such that $\frac{A G}{G B}=\frac{B H}{H C}=\frac{C I}{I A}=\rho$. For some choice of $\rho\gt 1$, $G H$, $H I$, and $I G$ are three edges of a regular icosahedron, eight of whose faces are inscribed in the faces of $A B C D E F$. Find $\rho$.