The configuration of seven circles and ten points
A proposition about hexahedra
The above proposition of inversive geometry can be translated to a statement about hexahedra. A cube is a six-faced polyhedron where each face is a square, and three squares meet at each vertex. Call any six-faced polyhedron where each face is a quadrilateral and three quadrilaterals meet at each vertex a hexahedron. A cube is a regular hexahedron, but there are plenty of irregular hexahedra.
The configuration of six circles and eight points in a plane can be projected to a sphere by means of the stereographic projection. And then the proposition holds for six circles on the sphere.
Each of these circles determines a plane, and the six planes determine the six faces of a hexahedron. The hexahedron so determined doesn't have to be a convex hexadron, but it could have intersecting faces. The proposition can now be stated in terms of hexahedra as follows: if seven of the eight vertices of a hexahedron lie on a sphere, then so does the eighth.
Special cases in Euclidean geometry
The configuration of six circles and eight points looks quite different when one of the points, say the point 0, is the point at infinity. Then three of the circles, A, B, and C are straight lines, and they form a triangle 123. The next three points, 4, 5, and 6, may be freely chosen on the three sides of that triangle. Let the next three circles A', B', and C' each pass through three of the points as before. Then the proposition states that these three circles all pass through the same point 7.
David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610
Email: djoyce@clarku.edu
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