  We give elementary proof of stronger versions of several recent results on
intrinsic Diophantine approximation on rational quadric hypersurfaces $X\subset
\mathbb{P}^n(\mathbb{R})$. The main tool is a refinement of the simplex lemma,
which essentially says that rational points on $X$ which are sufficiently close
to each other must lie on a totally isotropic rational subspace of $X$.
