Proof: place the spheres on a flat table, and put a board on top of them. The table and board define two planes that are tangent to all three spheres. The cones and their apexes also touch both planes. The apexes must lie on the intersection of these two planes, which is a line.

Reference

[1] L. A. Graham, Ingenious Mathematical Problems and Methods, New York: Dover, 1959 pp. 198–199.