This was an example of generalized pigeon-hole principle.
Ten dots are placed within a square of unit size. The textbook then shoes a box divided into 9 equal squares. Then there three dots that can be covered by a disk of radius 0.5.
Divide our square into four equal parts by it's diagonals (from one corner to the other), then by the generalized pigeon-hole principle, at least one of these triangles will contain three of our points. The proof follows as the radius of the circumcircle of these triangles is shorter than 0.5.
But wait! The statement said three dots can be covered by a disk of radius 0.5. Typo?