  Richmond and Richmond (American Mathematical Monthly 104 (1997), 713--719)
proved the following theorem: If, in a metric space with at least five points,
all triangles are degenerate, then the space is isometric to a subset of the
real line. We prove that the hypothesis is unnecessarily strong: In a metric
space on $n$ points, fewer than $7n^2/6$ suitably placed degenerate triangles
suffice. However, fewer than $n(n-1)/2$ degenerate triangles, no matter how
cleverly placed, never suffice.
