  We prove that if there exists a bi-Lipschitz homeomorphism (not necessarily
subanalytic) between two subanalytic sets, then their tangent cones are
bi-Lipschitz homeomorphic. As a consequence of this result, we show that any
Lipschitz regular complex analytic set, i.e any complex analytic set which is
locally bi-lipschitz homeomorphic to an Euclidean ball must be smooth. Finally,
we give an alternative proof of S. Koike and L. Paunescu's result about the
bi-Lipschitz invariance of directional dimensions of subanalytic sets.
