The co-yield surfaces in fcc single crystals for slip on {111}<110> and twinning on {111}<112> systems have been analyzed and derived systematically for the first time. The results demonstrated that only if the ratio xi of the critical resolved shear stress (CRSS) for twinning to that for slip is within the range of 1root3 less than or equal to xi less than or equal to 2/root, slip and twinning can occur together. There are only two types of co-yield surfaces when xi is within this range. The corresponding analytical expressions of all possible types of yield vertices are also derived and tabulated. One type consists of 259 co-yield stress states when root3/2 < ξ < 2/root3, which can be classified into 21 groups in stress space according to the crystal symmetry. The other type contains 259 co-yield stress states also when 1/root3 < ξ < root3/2, which can be subdivided into 19 groups. Of the two types of the co-yield stress states, 139 ones are common and 120 ones are different.