Is there a reference or a very short argument proving the following statement?

Let $C$ be a set consisting of $r$ points in the real projective space $\mathbb RP ^k$ with its usual round metric. Assume that the distances between all pairs of points in $C$ are the same. Assume further that $r>k+1$. Then $r$ must be equal to $k+2$ and $C$ must be the image of the set of vertices of a regular simplex inscribed in the unit sphere.