Puzzle 2: Solution

Q     Circles C and C' meet at points P and Q. A line tangent to both circles meets them at points A and A'. Consider the triangle T whose vertices are P, Q, and the midpoint of the line segment AA'. Given the radii r and r' of C and C', how large can the area of T be?

A     Zero. The three vertices of the “triangle” are always on the radical axis of C and C'!

Similarly, if C and C' are disjoint, there are four line segments tangent to both circles at their endpoints; the resulting four midpoints are collinear, the common line being again the radical axis.

Likewise in higher dimensions. For instance, three spheres close enough to intersect usually meet in two points, call them again P and Q, and have two planes tangent to all three spheres. For each of the planes, assume that the three points of tangency are not collinear, and consider the center of the unique circle through all three. Then P, Q, and the two circumcenters lie on the radical axis of the three spheres, and are thus collinear. If the spheres are disjoint, there can be as many as eight planes tangent to all three; this yields eight circumcenters, all on the radical axis and thus collinear.