Tipping a Tetrahedron
Source: 2011 MMO Problem #3
September 11, 2011
geometry3D geometrytetrahedronfrustumspheretrigonometryintegration
Problem Statement
A regular tetrahedron of height has a tetrahedron of height cut off by a plane parallel to the base. When the remaining frustrum is placed on one of its slant faces on a horizontal plane, it is just on the point of falling over. (In other words, when the remaining frustrum is placed on one of its slant faces on a horizontal plane, the projection of the center of gravity G of the frustrum is a point of the minor base of this slant face.)
Show that is a root of the equation .