Construction of the isothermal section in the metal-rich portion (<67 atom % P) of the Mo-Fe-P phase diagram at 800 degrees C has led to the identification of two new ternary phases: (Mo1-xFex)(2)P (x = 0.30-0.82) and (Mo1-xFex)(3)P (x = 0.10-0.15). The occurrence of a Co2Si-type ternary phase (Mo1-xFex)(2)P, which straddles the equiatomic composition MoFeP, is common to other ternary transition-metal phosphide systems. However, the ternary phase (Mo1-xFex)(3)P is unusual because it is distinct from the binary phase Mo3P, notwithstanding their similar compositions and structures. The relationship has been clarified through single-crystal X-ray diffraction studies on Mo3P (alpha-V3S-type, space group I<(4)over bar>2m, a = 9.7925(11) angstrom, c = 4.8246(6) angstrom) and (Mo0.85Fe0.15)(3)P (Ni3P-type, space group I (4) over bar, a = 9.6982(8) angstrom, c = 4.7590(4) angstrom) at -100 degrees C. Representation in terms of nets containing fused triangles provides a pathway to transform these closely related structures through twisting. Band structure calculations support the adoption of these structure types and the site preference of Fe atoms. Electrical resistivity measurements on (Mo0.85Fe0.15)(3)P reveal metallic behavior but no superconducting transition.