A few puzzles about sets of points, and a nice conjecture following up one of those questions:

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 presented 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?