Pigeon Hole Principle question, proving it is possible to select a number of points on a plane such that they are all interior to a circle

Question

Thirteen points are given in the plane so that among any three of them there is a pair whose distance apart is less than 1. Prove that it is possible to select seven of the points so that they are all interior to a circle of radius 1.

I've experimented with PHP, but I didn't get much out of it...

Just for the record, this isn't a homework question, I stumbled upon it when I was doing some other PHP questions (no solution was given :( ...)

Answer 1

Hint: Pick any point ($A$) and consider the circle of radius 1 about $A$.

Case 1: All points lie in the circle, we are done.

Case 2: There is some point $B$ that doesn't lie in this circle.

Claim: All 13 points lie in a circle of radius 1 about A or about B.

Corollary: Hence, by PP, we are done.