## Points on a plane

A few puzzles about sets of points:

Suppose $S$ is a finite set of points on a plane, such that for any two points $A,B \in S$, there is a third point $D \in S$, collinear with $A$ and $B$ (and different from $A$ and $B$).

Show that all points of $\boldsymbol{S}$ are collinear.

The nicest solution I know of this problem comes by reductio ad absurdum, and goes like this:

Suppose that not all points in $S$ are collinear. Among all possible triangles that we can form using points from this set as vertices, chose the one with the smallest height, say $\delta$; label the vertices at the base of the triangle $A$ and $B$, and the remaining vertex $C.$ It will not be hard for the reader to realize that this implies, in particular, that the projection of $C$ on the line from $A$ and $B$ is either one of the previous two vertices, or it is located inside of the segment that joins them.

By hypothesis, there is a third point $D$ which is collinear with $A$ and $B$. Consider now the new triangles that arose: $\triangle ADC, \triangle BDC.$ The trick now is to realize that the smallest of the heights of one of these two triangles, say $\delta',$ is necessarily smaller than $\delta$. This is a contradiction!

The second puzzle was sent by Ralph Howard:

Let $S$ be a infinite set of points on a plane, such that the distance between any two is an integer.

Prove that all points in $\boldsymbol{S}$ are collinear.

Ralph pointed up to a clever solution to this riddle by Paul Erdös. He published it in a short article, titled Integral distances, published in Bull. Amer. Math. Soc. 51, (1945). 996. It goes as follows:

Suppose you have three points $A$, $B$ and $C$ with integer distances between them and not all on the same line. Let us denote $\mathop{d}(A,B)$ the distance between points $A$ and $B$. If $\mathop{d}(A,Q)$ and $\mathop{d}(B,Q)$ are both integers, note that $\mathop{d}(A,Q) - \mathop{d}(B,Q)$ is one of the integers in the closed interval $[-\mathop{d}(A,B), \mathop{d}(A,B)].$ Now for any given integer $k$, the points $Q$ satisfying $\mathop{d}(A,Q) - \mathop{d}(B,Q) = k$ all lie on a branch of a hyperbola (or the degenerate cases: a straight line parallel or perpendicular to the line through $A$ and $B$). Every point of the set $S$ is an intersection of one of these curves, and one of the analogous curves for $A$ and $C$, and one of the curves for $B$ and $C$. But any two of the curves intersect in only a finite number of points. Therefore there are only a finite number of points with integer distances from $A$, $B$ and $C.$ This is a contradiction!

Ralph conjectures too that the same result holds if the points lie in any other higher-dimension space:

I conjecture that the same is true in three and higher dimensions. That is: if $S$ is an infinite set of points in $\mathbb{R}^d$ such that the distance between any two is an integer, then the points of $S$ are colinear.

How would you prove this conjecture?