MathDB
concurrence of four planes

Source: Vietnam NMO 1995, problem 4

September 7, 2008
geometry3D geometrytetrahedroncircumcirclegeometry proposed

Problem Statement

Let a tetrahedron ABCD ABCD and A′,B′,C′,D′ A',B',C',D' be the circumcenters of triangles BCD,CDA,DAB,ABC BCD,CDA,DAB,ABC respectively. Denote planes (PA),(PB),(PC),(PD) (P_A),(P_B),(P_C),(P_D) be the planes which pass through A,B,C,D A,B,C,D and perpendicular to C′D′,D′A′,A′B′,B′C′ C'D',D'A',A'B',B'C' respectively. Prove that these planes have a common point called I. I. If P P is the center of the circumsphere of the tetrahedron, must this tetrahedron be regular?
Back to ProblemsView on AoPS