Stereographic Projection of a Coffin Problem

Hubert Shutrick came up with an elegant solution to one of the coffin problems:

So let there be a 3D quadrilateral A₁A₂A₃A₄ with the property that all of its 4 sides, A₁A₂, A₂A₃, A₃A₄, A₄A₁, are tangent to a given sphere. (The side A_i_j touches the sphere at point T_ij.) Interestingly, the four points of tangency T₁₂, T₂₃, T₃₄, T₄₁ are necessarily coplanar.

One solution based on the idea of center of gravity (barycenter) appeared elsewhere.

Hubert Shutrick's solution draws on the conformality (angle preservation) propety of the stereographic projection.

Solution

Stereographic projection converts the coffin problem into a problem in Euclidean geometry that is interesting in its own right. Since each vertex of the skew quadrilateral along with two adjacent tangency points defines a circle on the sphere by the plane of the sides that meet in the vertex; these four circles A, B, C, D can be projected down onto the plane and A is tangent to B is tangent to C is tangent to D is tangent to A (because, any successive two share a point but lie in different planes). Show that the points where they touch lie on a circle. It is not difficult to prove by just chasing angles but, of course, follows directly from the elegant solution of the original problem.

Stereographic Projection

Stereographic Projection and Radical Axes
Stereographic Projection and Inversion
Stereographic Projection of a Coffin Problem
Stereographic Projection - an Interactive Tool
Objects distant and near

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