HMMT 二月 2006 · 几何 · 第 9 题

HMMT February 2006 — Geometry — Problem 9

Four spheres, each of radius r , lie inside a regular tetrahedron with side length 1 such that each sphere is tangent to three faces of the tetrahedron and to the other three spheres. Find r .

解析

Four spheres, each of radius r , lie inside a regular tetrahedron with side length 1 such that each sphere is tangent to three faces of the tetrahedron and to the other three spheres. Find r . 2 √ 6 − 1 Answer: 10 Solution: Let O be the center of the sphere that is tangent to the faces ABC , ABD , and BCD . Let P , Q be the feet of the perpendiculars from O to ABC and ABD respectively. Let R be the foot of the perpendicular from P to AB . Then, OP RQ is a quadrilateral such that ∠ P , ∠ Q are right angles and OP = OQ = r . Also, ∠ R is the dihedral angle between faces ABC and ABD , so cos ∠ R = 1 / 3 . We can then √ √ √ √ compute QR = 2 r , so BR = 6 r . Hence, 1 = AB = 2( 6 r ) + 2 r = 2 r ( 6 + 1), so √ r = ( 6 − 1) / 10 .