Now, a plane tangent to a cone necessarily contains its vertex. Therefore, planes #1 and #2 are tangent to each of the three cones. Any plane (externally) tangent to two spheres will also be tangent to the enclosing cone. On the other hand, any two of the spheres can be inscribed into a cone. By definition, plane #3 passes through l. This will be the plane, plane #3, depicted at the top of the page. The bisector plane of the solid angle formed by planes #1 and #2 passes through the centers of all three spheres. For then planes #1 and #2 are bound to have a common line l, the line of their intersection. In one particular case where there exists a second plane, plane #2, tangent to the three spheres but covering them from above the proof is particularly intuitive. Imagine having three different spheres lying on a plane, say plane #1.

The following is the argment put forth by John Edson Sweet.

|Contact| |Front page| |Contents| |Geometry| |Up|Ĭopyright © 1996-2018 Alexander Bogomolny D.Wells tells of John Edson Sweet, an engineer, whose reaction to the above diagram was, 'Yes, that is perfectly self-evident.' Solutions It has a '3D solution' but can also be solved without leaving the plane. This problem is almost on a par with the 4 Travelers and 9 dots problems. Under these conditions, six external tangents to two of the three circles, taken pairwise, intersect at three points. the three centers do not lie on the same straight line. To exclude a trivial case, assume also that their centers are not collinear, i.e. Let there be three circles of different radii lying completely outside each other.

Where Tangents Meet by NOT A BOOK