A new numerical method called "Homotopy" method (Continuation method) is applied to the problem of four-bar coupler-point-curve synthesis. We have shown that, for five precision points, the link lengths of a four-bar lilcage can be found by the "General Homotopy" method. For nine precision points, the "Cheater's Homotopy" can be applied to find some fourbar linkages that will guide a couDler noint through the nine prescribed positions. The nine-coupler-points synthesis problem is highly non-linear and highly singular. We have also shown that NewtonRaphson's method and Powell's method, in general, tend to converge to the singular condition or do not converge at all, while the Cheater's homotopy always works. The powerfulness of Cheater's homotopy opens a new frontier for dimensional synthesis of mechanisms.