Symmetry-adapted perturbation theory (SAPT) has been used to analyze the radial and angular dependence of the nonadditivity of the Ar2HF trimer interaction energy through fourth order. This represents the first application of the high-order SAPT to a nonadditive interaction including a polar molecule. The magnitude and anisotropy of the Hartree-Fock nonadditivity is well reproduced (to within 20%) by the sum of the first-order exchange and exchange-quenched third-order induction nonadditivities. The second-order induction effects play a smaller role. The computed SAPT corrections which contribute to the second-order supermolecular many-body perturbation theory (MBPT2) nonadditivity, E-exch-disp((2;0))[3,3] and E-ind-disp((3;0))[3,3], reproduce MBPT2 values rather poorly. Using the pseudo-dimer approach it was found that the exchange quenching of the third-order induction-dispersion energy is strong. Inclusion of this quenching led to good agreement with the MBPT2 nonadditivity. The third-order MBPT nonadditivity was very well reproduced by the third-order dispersion energy. The fourth-order MBPT nonadditivity was only moderately well reproduced by the SAPT components E-disp((3;1))[3,3] and E-disp((4;0))[3,3], indicating that these terms are most likely appreciably quenched by exchange counterparts. The total nonadditivities computed using SAPT and the supermolecular method through fourth order agree remarkably well. The total SAPT nonadditivity is expressed in terms of physically interpretable components which can be easily modeled.