Ground state properties of the spin$-1/2$ Falicov-Kimball model on a triangular lattice in the presence of uniform external magnetic field are explored. Both the orbital and the Zeeman field-induced effects are taken into account and in each unit cell only rational flux fractions are considered. Numerical results, obtained with the help of Monte Carlo simulation algorithm, reveal that the ground state properties strongly depend on the onsite Coulomb correlation between itinerant and localized electrons, orbital magnetic field as well as the Zeeman splitting. Strikingly, for the on-site Coulomb correlation $U/t approx 1$, the Zeeman splitting produces a phase transition from paramagnetic metal/insulator to ferromagnetic insulator/metal transition in the itinerant electron subsystem accompanied by the phase segregation to the bounded/regular phase in the localized electrons subsystem. For the onsite Coulomb correlation $U/t approx 5$, although no metal to insulator transition is observed but a magnetic phase transition from paramagnetic phase to ferromagnetic phase in the itinerant electron subsystem is observed with the Zeeman splitting. These results are applicable to the layered systems e.g. cobaltates, rare earth and transition metal dichalcogenides, $GdI_{2}$, $NaTiO_{2}$, $NaVO_{2}$ and $Be_{x}Zn_{1-x}O$ etc. It has been also proposed that the results can be realized in the optical lattices with mixtures of light atoms and heavy atoms using the cold atomic techniques.
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