  We present an algorithm for computing circuit polynomials in the algebraic
rigidity matroid $\mathcal{A}(\text{CM}_n)$ associated to the Cayley-Menger
ideal CM$_n$ for $n$ points in 2D. It relies on combinatorial resultants, a new
operation on graphs that captures properties of the Sylvester resultant of two
polynomials in this ideal. We show that every rigidity circuit has a
construction tree from K4 graphs based on this operation. Our algorithm
performs an algebraic elimination guided by such a construction tree, and uses
classical resultants, factorization and ideal membership. To highlight its
effectiveness, we implemented the algorithm in Mathematica: it took less than
15 seconds on an example where a Gr\"obner Basis calculation took 5 days and 6
hrs. Additional speed-ups are obtained using non-$K_4$ generators of the
Cayley-Menger ideal and simple variations on our main algorithm.
