We shall discuss the properties of compact minimal Lagrangian submanifolds in a complex hyperquadric obtained as the Gauss images of isoparametric hypersurfaces in a sphere. Our main results are as follows: (a) The Gauss image is a monotone and cyclic Lagrangian submanifold in a complex hyperquadric with minimla Maslov number $2n/g$, and it is orientable if and only if $2n/g$ is even. (b) A complete classification of compact homogeneous Lagrangian submanifolds in complex hyperquadrics. (c) The determination of the (strictly) Hamiltonian stability of the Gauss images of all compact homogeneous isoparametric hypersurfaces in spheres. This talk is based on my joint works with Hui Ma (Tsinghua University, Beijing, China).