The discrete Steiner point is a valuation on lattice polytopes, defined by its equivariance under translations and unimodular transformations, together with the valuation property. This talk introduces its fundamental properties and examines how it behaves under standard geometric operations. The main focus lies on analysing families of lattice triangles with fixed base and varying apex. The discrete Steiner point is computed for many such examples, and closed-form expressions for its coordinates are derived using interpolation techniques and moment formulas. Additionally, we explore how the point behaves under triangulations, Minkowski sums, and symmetry, and how these methods can be used to extend computations to more complex shapes.