IMO 1985 SL 9

Problem

2a.(USA 3) Determine the radius of a sphere S that passes through the centroids of each face of a given tetrahedron T inscribed in a unit sphere with center O. Also, determine the distance from O to the center of S as a function of the edges of T .

Solution

Let −\toa , −\tob , −\toc , −\tod denote the vectors −\to OA, −−\to OB, −−\to OC, −−\to OD respectively. Then |−\toa | = |−\tob | = |−\toc | = |−\tod | = 1. The centroids of the faces are (−\tob +−\toc +−\tod )/3, (−\toa + −\toc + −\tod )/3, etc., and each of these is at distance 1/3 from P = (−\toa + −\tob + −\toc + −\tod )/3; hence the required radius is 1/3. To compute |P| as a function of the edges of ABCD, observe that AB2 = (−\tob −−\toa )2 = 2 −2−\toa \cdot −\tob etc. Now P 2 = |−\toa + −\tob + −\toc + −\tod |2 = 16 −2(AB2 + BC2 + AC2 + AD2 + BD2 + CD2) .