Nonadditivity contribution to the three-body interaction energy is studied in terms of the symmetry-adapted perturbation theory for many-electron systems. Each component of energy, particularly the second-order exchange dispersion and exchange induction contributions, is given by a combination of electrostatic interaction energies in Longuet-Higgins representation of the intermolecular charge distribution. The formulas of these energies are derived with the Hartree-Fock approximation and by taking triple-electronic exchanges among three monomers into account. Numerical calculation has been performed for the cyclic planar H2O and HF trimers, considering only single-electronic exchanges between molecules. The three-body effect of the second-order exchange energy has been found to be repulsive, while the main part of attractive contribution is due to the induction. The ratio of three-body energy to two-body one for the dispersion is much smaller than that for the induction, though the latter decreases more rapidly than the former as the angle between monomers increases. As a result, the three-body contribution lowers the total interaction energy slightly near the van der Waals minimum in both trimers. The contribution of two-body energies is also shown.