Call a number abnormal if its decimal expansion doesn't feature every digit an infinite number of times. Call a triangle in ${\Bbb R}^2$ abnormal if at least one of its angles spans an abnormal fraction of $2\pi$ radians.

Assuming the Continuum Hypothesis, by the obvious transfinite induction one gets a set $M$ so small that no three of its points form an abnormal triangle, but so large that every point of ${\Bbb R}^2$ either falls in $M$ or else occurs as the midpoint of two points in $M$.

Does such an $M$ exist in ZFC?