This paper proposes a novel formulation to the problem of the five-precision points path synthesis for planar four-bar linkages with given fixed pivots. Firstly, the coupler-curve equation of planar four-bar linkages is revisited and derived based on the conformal geometric algebra (CGA), whose derivation is operated geometrically and free of coordinate. Secondly, based on the derived coupler-curve equation and the variable substitution, the seven constraint equations in seven variables for the problem are formulated. Thirdly, the Gröbner bases under the group degree reverse lexicographic ordering for the problem are reduced using computer algebra. Then, a 19 by 19 Sylvester resultant is constructed by selecting 19 Gröbner bases from 48 ones. Finally, a 36th degree univariate equation is directly obtained from the determinant of the resultant. At last, two numerical examples are provided to demonstrate the proposed method. The advantages of the novel method lie in that seven constraint equations are derived based on the coupler curve, and the novel formulation is concise and the computing speed is faster than before.