A sagittal-plane model of the knee is used to predict and explain the relationships between the forces developed by the muscles, the external loads applied to the leg, and the forces induced in the cruciate ligaments during isometric exercises. The geometry of the model bones is adapted from cadaver data. Eleven elastic elements describe the geometric and mechanical properties of the cruciate ligaments, the collateral ligaments, and the posterior capsule. The model is actuated by 11 musculotendinous units, each unit represented as a three-element muscle in series with tendon. For isolated contractions of the quadriceps, ACL force increases as quadriceps force increases for all flexion angles between 0 and 80 degrees; the ACL is unloaded at flexion angles greater than 80 degrees. When quadriceps force is held constant, ACL force decreases monotonically as knee-flexion angle increases. The relationship between ACL force, quadriceps force, and knee-flexion angle is explained by the geometry of the knee-extensor mechanism and by the changing orientation of the ACL in the sagittal plane. For isolated contractions of the hamstrings, PCL force increases as hamstrings force increases for all flexion angles greater than 10 degrees; the PCL is unloaded at flexion angles less than 10 degrees. When hamstrings force is held constant, PCL force increases monotonically with increasing knee flexion. The relationship between PCL force, hamstrings force, and knee-flexion angle is explained by the geometry of the hamstrings and by the changing orientation of the PCL in the sagittal plane. At nearly all knee-flexion angles, hamstrings co-contraction is an effective means of reducing ACL force. Hamstrings co-contraction cannot protect the ACL near full extension of the knee because these muscles meet the tibia at small angles near full extension, and so cannot apply a sufficiently large posterior shear force to the leg. Moving the restraining force closer to the knee-flexion axis decreases ACL force; varying the orientation of the restraining force has only a small effect on cruciate-ligament loading.