We will say that a set $ A $ of points is disastrous if it meets the following conditions: $\bullet$ There are no $ 3 $ collinear points $\bullet$ There is not a trio of mutually equal distances between points. If $ P $ and $ Q $ are points in $ A $, then there are $ M $, $ N $, $ R $ and $ T $ in $ A $ such that: $$ d (P, Q) = \frac {d (M, N) + d (R, T)} {2} $$Show that all disastrous sets are infinite.

HIDE: original wording of second condition No existe ni un trío de distancias entre puntos mutuamente iguales.