# Pigeon-hole Principle: Does this proof have a typo?

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.

The proof:

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?

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The proof is basically correct, but yes, there is a typo: the circumcircle of each of the four triangles has radius exactly $0.5$, not less than $0.5$. If $O$ is the centre of the square, and $A$ and $B$ are adjacent corners, the centre of the circumcircle of $\triangle AOB$ is the midpoint of $\overline{AB}$, from which the distance to each of $A,O$, and $B$ is $0.5$. The circle of radius $0.5$ and centre at the midpoint of $\overline{AB}$ contains $\triangle AOB$, as required.
The proof is correct. The furthest away from each other the points can be is in the three vertices of the $\sqrt 2/2,\ \sqrt 2/2,\ 1$ right triangle. This triangle has circumradius 0.5, so all three points are still covered by a disk with radius 0.5.