Prove that there is a unique point satisfying conditions
Source: Vietnam TST 1997, Problem 1
July 28, 2008
geometry3D geometrytetrahedroncircumcircleinequalitiesanalytic geometrygeometry unsolved
Problem Statement
Let be a given tetrahedron, with BC \equal{} a, CA \equal{} b, AB \equal{} c, DA \equal{} a_1, DB \equal{} b_1, DC \equal{} c_1. Prove that there is a unique point satisfying
PA^2 \plus{} a_1^2 \plus{} b^2 \plus{} c^2 \equal{} PB^2 \plus{} b_1^2 \plus{} c^2 \plus{} a^2 \equal{} PC^2 \plus{} c_1^2 \plus{} a^2 \plus{} b^2 \equal{} PD^2 \plus{} a_1^2 \plus{} b_1^2 \plus{} c_1^2
and for this point we have PA^2 \plus{} PB^2 \plus{} PC^2 \plus{} PD^2 \ge 4R^2, where is the circumradius of the tetrahedron . Find the necessary and sufficient condition so that this inequality is an equality.