  An astroid-shaped loop of exceptional points (EPs), comprising four cusps, is
found to spawn from the triple degeneracy point in the Brillouin zone (BZ) of a
Lieb lattice with nearest-neighbor hoppings when non-Hermiticity is introduced.
The occurrence of the EP loop is due to the realness of the discriminant which
is guaranteed by the non-Hermitian chiral symmetry. The EPs at the four cusps
involve the coalescence of three eigenstates, which is the combined result of
the non-Hermitian chiral symmetry and mirror-T symmetry. The EP loop is exactly
an astroid in the limit of an infinitesimal non-Hermiticity. The EP loop
expands from the $M$ point with increasing non-Hermiticity and splits into two
EP loops at a critical non-Hermiticity. The further increase of non-Hermiticity
contracts the two EP loops towards and finally to two EPs at the $X$ and $Y$
points in the BZ, accompanied by the emergence of Dirac-like cones. The two EPs
vanish at a larger non-Hermiticity. The EP loop disappears and several discrete
EPs are found to survive when next-nearest hoppings are introduced to break the
non-Hermitian chiral symmetry. A topological invariant called the discriminant
number is used to characterize their robustness against perturbations. Both
discrete EPs and those on the EP loop(s) are found to show anisotropic
asymptotic behaviors. Finally, the experimental realization of the Lieb lattice
using a coupled waveguide array is discussed.
