Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.2 AIME, harder Find the answer

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 $\rho>1, G H, H I$ , 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 $\rho$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Let $J$ lie on edge $C E$ such that \frac{E J}{J C}=\rho. Then we must have that $H I J$ is another face of the icosahedron, so in particular, $H I=H J$ . But since $B C$ and $C E$ are perpendicular, $H J=H C \sqrt{2}$ . By the Law of Cosines, $H I^{2}=H C^{2}+C I^{2}-2 H C \cdot C I \cos 60^{\circ}=$ $H C^{2}\left(1+\rho^{2}-\rho\right)$ . Therefore, $2=1+\rho^{2}-\rho$ , or $\rho^{2}-\rho-1=0$ , giving $\rho=\frac{1+\sqrt{5}}{2}$ .

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