58. Three spheres with different radii and centers not lying on the same straight line lie outside each other. What figure is formed by the lines of intersection of pairs of planes that are symmetric with respect to the plane of the centers of the spheres and simultaneously tangent to all three given spheres?
58. The common tangent plane of two spheres passes through the center of their homothety, since the radii drawn to the points of tangency are parallel, and the line passing through the points of tangency also passes through the center of homothety (see the previous problem). The centers of homothety of three spheres belong to the plane of their centers and coincide with the centers of homothety of the circles of section of the spheres by this plane. Therefore, the plane tangent to all three spheres passes through the axes of homothety of these circles (see problem 15 from §2).

Since the spheres are symmetric relative to the plane of their centers, through each axis of homothety there pass pairs of mutually symmetric planes tangent to the three spheres.

Thus, pairs of symmetric planes tangent to three spheres intersect along lines lying in the plane of the centers and form a configuration of 4 lines and 6 points, with 3 points on each line.