  We investigate the question of how well points on a nondegenerate
$k$-dimensional submanifold $M \subseteq \mathbb R^d$ can be approximated by
rationals also lying on $M$, establishing an upper bound on the "intrinsic
Dirichlet exponent" for $M$. We show that relative to this exponent, the set of
badly intrinsically approximable points is of full dimension and the set of
very well intrinsically approximable points is of zero measure. Our bound on
the intrinsic Dirichlet exponent is phrased in terms of an explicit function of
$k$ and $d$ which does not seem to have appeared in the literature previously.
It is shown to be optimal for several particular cases. The requirement that
the rationals lie on $M$ distinguishes this question from the more common
context of (ambient) Diophantine approximation on manifolds, and necessitates
the development of new techniques. Our main tool is an analogue of the Simplex
Lemma for rationals lying on $M$ which provides new insights on the local
distribution of rational points on nondegenerate manifolds.
