congruent tetrahedra
Source: Polish MO Recond Round 1978 p5
September 9, 2024
3D geometrygeometrytetrahedron
Problem Statement
Prove that there is no inclined plane such that any tetrahedron placed arbitrarily with a certain face on the plane will not fall over. It means the following:Given a plane and a line not perpendicular to it. Prove that there is a tetrahedron such that for each of its faces there is in the plane a triangle congruent to and there is a point such that the tetrahedron congruent to and the line parallel to passing through the center of gravity of the tetrahedron does not intersect the triangle .Note. The center of gravity of a tetrahedron is the intersection point of the segments connecting the centers of gravity of the faces of this tetrahedron with the opposite vertices (it is known that such a point always exists).