  The magnetic phase diagram and the thermodynamic properties of a square
lattice containing antiferromagnetically coupled spin-1/2 triangles are
studied. A Heisenberg Hamiltonian with three strong intra-triangle exchange
coupling constants and one weak inter-triangle exchange coupling constant is
proposed to model the interacting Cu^2+ ions of the Cu_3MoO_4 planes in
La_4Cu_3MoO_12. Depending on the ratio of the intra-triangle coupling
constants, various long-ranged magnetic phases are shown to compete at low
temperatures. A comparison of numerical calculations of thermodynamic
properties in the model Hamiltonian with recent experiments on La_4Cu_3MoO_12
suggests that the spins in this material order antiferromagnetically along the
x-direction and ferromagnetically along the y-direction of the Cu_3MoO_4
planes. The temperature dependence of the uniform magnetic susceptibility, the
heat capacity, and the entropy are calculated and are shown to be in
qualitative agreement with the experiments on this compound.
