7 points on a closed disk, one must be the center

Question

I am trying a pigeonhole strategy for proving this assertion of a MO test:

"Given 7 points on a closed disk of radius 1 such that the distance between any two of this points is at least one, then one of this points is the center of the disk."

As always in this kind of problem, the difficult seems to be in applying the principle in the right way. Could someone give a hint (and not a full solution) for this puzzle?

Answer 1

Hint: Inscribe a regular hexagon inside the circle and then show that by the pigeonhole principle there must be at least $2$ points in the same sector.