We study random triangulations of the integer points [0,n]^2 cap {mathbb {Z}}^2, where each triangulation sigma has probability measure lambda ^{|sigma |} with lambda >0 being a real parameter and |sigma | denoting the sum of the length of the edges in sigma . Such triangulations are called lattice triangulations. We construct a height function on lattice triangulations and prove that, in the whole subcritical regime lambda <1, the function behaves as a Lyapunov function with respect to Glauber dynamics; that is, the function is a supermartingale. We show the applicability of the above result by establishing several features of lattice triangulations, such as tightness of local measures, exponential tail of edge lengths, crossings of small triangles, and decay of correlations in thin rectangles. These are the first results on lattice triangulations that are valid in the whole subcritical regime lambda <1. In a very recent work with Caputo, Martinelli and Sinclair, we apply this Lyapunov function to establish tight bounds on the mixing time of Glauber dynamics in thin rectangles that hold for all lambda <1. The Lyapunov function result here holds in great generality; it holds for triangulations of general lattice polygons (instead of the [0,n]^2 square) and also in the presence of arbitrary constraint edges.