Track / Stage 5 / 78 of 400 #678 of 1964

Problem 678

AIME late

Geometry Difficulty 5.1 Find the answer

The lateral faces of a triangular pyramid are equal in area and form angles $\alpha, \beta$ and $\gamma$ with the base. Find the ratio of the radius of the sphere inscribed in this pyramid to the radius of the sphere that touches the base of the pyramid and the extensions of the lateral faces.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

3.38. Let $r$ and $r^{\prime}$ be the radii of the inscribed and exscribed spheres, $S$ the area of the lateral face, $s$ the area of the base, and $V$ the volume of the pyramid. Then $V=(3 S+s) r / 3$ . Similarly, it can be shown that $V=(3 S-s) r^{\prime} / 3$ . Moreover, $s=(\cos \alpha + \cos \beta + \cos \gamma) S$ (see problem 2.13). Therefore,

$\frac{r}{r^{\prime}}=\frac{3 S-s}{3 S+s}=\frac{3-\cos \alpha-\cos \beta-\cos \gamma}{3+\cos \alpha+\cos \beta+\cos \gamma}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.