The large number of above-knee amputations world-wide creates a need for prosthetic knees with kinematic characteristics fit to individual users. Prosthetists use a number of parameters to fit an individual: leg length in zero and 65 degree flexion, cosmetic advantage in 90 degree flexion, and location of the coupler instant center of rotation both above the knee joint and behind its axis in the first position. Designers use four-bar and similar mechanisms to provide acceptable motion. The first three design requirements point to three-position motion synthesis, a well-studied kinematic synthesis approach for which a four-bar designer exercises two free choices. However, specification of the instant center location adds constraints which may force the designer to iterate or compromise in order to satisfy space and other constraints. The three-position, instant-center specification problem is generally applicable to a number of other situations. It is shown that the addition of the instant center requirement reduces the number of free choices available to the designer to only one, so that pivot locations may be located along a locus similar to a Burmester curve. Results of work to produce such curves graphically and computationally is presented.