The purpose of this paper is to present a new method to design exact geometric predicates in algorithms dealing with curved objects such as circular arcs. We focus on the comparison of the abscissae of two intersection points of circle arcs, which is known to be a difficult predicate involved in the computation of arrangements of circle arcs. We present an algorithm for deciding the $x$-order of intersections from the signs of the coefficients of a polynomial, obtained by a general approach based on resultants. This method allows the use of efficient arithmetic and filtering techniques leading to fast implementation as shown by the experimental results.