Prove/disprove the Lemma
$Lemma$ : There are 12 distinct vertices (Points) in $\mathbb{R^3}$. Given any vertex the minimum distance to another vertex is $1$ $unit$ and there are exactly 5 such neighboring vertices with the distance $1$ $unit$ from the given vertex. Any such 12 vertices form an Icosahedron.
Obviously it should be shown that any such 12 vertices form equilateral triangles of the Icosahedron. But I am unable to come up with any argument for the same.
All Help will be appreciated.