Principle of Mathematical Induction

(Berkeley Math Tournament 2016, Geometry Focus #7, modified) A regular unit $n$-simplex is a polytope in $n$-dimensional space with $n+1$ vertices that are all exactly a distance of $1$ apart. (It is the $n$-dimensional analogue to the triangle and the tetrahedron.) In this $n$-dimensional space, there exists a point that is equidistant from all $n+1$ vertices, at a distance $d.$ Determine $d$ in terms of $n.$