MathDB

tetrahedron, barycenter / incenter / circumcenter statements equivalent

Source: France 1992 P3

May 13, 2021

3D geometrygeometry

Problem Statement

Let

ABCD

be a tetrahedron inscribed in a sphere with center

O

, and

G

and

I

be its barycenter and incenter respectively. Prove that the following are equivalent: (i) Points

O

and

G

coincide. (ii) The four faces of the tetrahedron are congruent. (iii) Points

O

and

I

coincide.