In this paper, we introduce the notion of strong geometry, a structure composed by both the chirotope of a set of points X in the d-dimensional space and the wedge chirotope which is the specific adjoint chirotope induced by the hyperplanes spanned by X. We present various properties relating these two chirotopes, for instance, by introducing the witness chirotope, we are able to give a formula expressing the wedge chirotope in terms of the usual chirotope. With this on hand, we answer positively a strong geometry version of a question due to M. Las Vergnas about reconstructing polygonal knots via chirotopes. Moreover, we also show that linear spatial graphs are determined by their corresponding strong geometries.