HMMT 二月 2006 · 几何 · 第 9 题

英文解析

9. Four spheres, each of radius r , lie inside a regular tetrahedron with side length 1 suchthat each sphere is tangent to three faces of the tetrahedron and to the other threespheres. Find r .
   √2
   6 − 1
   Answer:
   Solution: Let O be the center of the sphere that is tangent to the faces ABC , ABD ,10
   and BCD . Let P , Q be the feet of the perpendiculars from O to ABC and ABDrespectively. Let R be the foot of the perpendicular from P to AB . Then, OP RQis a quadrilateral such that ∠ P , ∠ Q are right angles and OP = OQ = r . Also, ∠ Ris 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 .