Track / Stage 4 / 145 of 340 #405 of 2000

Problem 405

AMC 12 late, AIME early

Geometry Difficulty 4.7 Find the answer

4.2. Three spheres touch each other pairwise; a plane touches these spheres at points $A, B$ , and $C$ . Find the radii of the spheres if the sides of triangle $ABC$ are equal to $a, b$ , and $c$ .

Your answer

Official solution

4.2. Let $x, y$ and $z$ be the radii of the spheres. According to problem 4.1, $a=2 \sqrt{x y}, b=2 \sqrt{y z}$ and $c=2 \sqrt{x z}$ . Therefore, $a c / b=2 x$ , hence $x=a c / 2 b$ . Similarly, $y=a b / 2 c$ and $z=b c / 2 a$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.