A conjecture on line transversals to five unit discs

In this paper, we prove the following: Let F = { x 1 + B , x 2 + B , x 3 + B , x 4 + B , x 5 + B } be a family of translates of the unit diameter disc B such that every three members of F are intersected by some straight line, then there is a line intersecting every member of the family F ′ = { x 1 + φ B , x 2 + φ B , x 3 + φ B , x 4 + φ B , x 5 + φ B } , where φ = 1 + 5 2 .