The combinatorial resultant is an abstraction of the classical Sylvester resultant for variable elimination among two multi-variate polynomials. As a graph operation, it leads to inductive constructions for rigidity-relevant family of sparse graphs, specifically the circuits of the 2D rigidity matroid. In the algebraic rigidity matroid induced by the 2D Cayley-Menger ideal, it is applied to the K4-generators of the ideal to guide the computation of circuit polynomials. I will do an overview of these result along with generalizations, implications and open problems. This is join work with Goran Malic.