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 Gröbner 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.