Choose distinct real parameters for the vertices and use the rational parametrization of the unit circle. A tangent to the circle at any vertex strictly separates that vertex from every chord with different endpoints. All edges of the star contain its centre, so there are no vertex-disjoint edges whose segments could cross. The construction includes the one-vertex star.

Direct elementary circle construction. Diestel, Graph Theory, sixth edition, Chapter 4, supplies the plane-drawing framework and the outerplanarity definition in Exercise 23, but does not give this explicit parametrization. The coordinate proof here is supplied in full and uses no planarity theorem.