  We investigate vertices for plane curves with singular points. As plane
curves with singular points, we consider Legendre curves (respectively,
Legendre immersions) in the unit tangent bundle over the Euclidean plane and
frontals (respectively, fronts) in the Euclidean plane. We define a vertex
using evolutes of frontals. After that we define a vertex of a frontal in the
general case. It is also known that the four vertex theorem does not hold for
simple closed fronts. We give conditions under which a frontal has a vertex and
the four vertex theorem holds for closed frontals. We also give examples and
counter examples of the four vertex theorem.
