Given a convex polygon $P=(P$_{1},P_{2},...,P_{n}) on a halfsphere, consider the planes
determined by the edges of the polygon and the center O of the sphere. Associate to each
of these planes, which can be represented as planes tangent to the cone determined by the
largest inscribed circle and the center of the sphere, its polar point $Q$_{i} that is the
intersection of the sphere with the external normal ray to the plane through the origin.

Any circle C on the half sphere defines a cone A that has a polar cone A* that
intersects the sphere circle C is inscribed in the polygon $P=(P$_{1},P_{2},...,P_{n}) if the polar
circle C* encloses points $Q$_{1},Q_{2},...,Q_{n}. The relation between the aperture angles of cones a* = p - d allows to reduce the problem of
finding the largest circle C inscribed in polygon P to the problem of finding the smallest
circle C* containing $Q$_{1},Q_{2},...,Q_{n}, that can be determined by the point belonging to the
polyhedral region which minimizes the distance to the center of the halfsphere,
considering the segment line that joins it to the center and intersecting it with the
halfsphere.

Let us start looking at the two dimensional case. Consider n points $Q$_{1},Q_{2},...,Q_{n} on
a halfcircle.

The halfplanes defined by the lines through $Q$_{1},Q_{2},...,Q_{n} tangent to the circle have
an intersection point N, that represents a vertex of the polygonal region and minimizes
the distance to the center of the circle. That is why the fact that points lie on a
halfcircle are crucial. All the halfplanes are redundant except two of them corresponding
to the first and the last point. The center K of the minimum spanning arc of the
points is the intersection of the segment line and the line through the points O and N,
since the two triangles $OQ$_{1}N and $OQ$_{n}N are congruent., the line ON bisects the minimum arc
containing the points.

The three dimensional case is analogous. The polyhedral region is formed by the
planes tangent to the halfsphere the points $Q$_{1},Q_{2},...,Q_{n} ( polar to $P$_{1},P_{2},...,P_{n}). The
region is a pyramid because all the planes intersect in an unique point or vertex. The
vertex of the pyramid is the same as the vertex of the tangent cone determined by the
minimum spanning circle because the planes forming the pyramid are tangent to the sphere
at the points belonging to the boundary of the minimum covering cap and these are tangent
to the cone. Therefore, the center of the minimum spanning circle, the center of the
sphere and the vertex of the polyhedral region, that minimizes the distance to the center
of the sphere, lie on a line.