This paper presents a new proof to a polynomial-time algorithm for determining whether a given embedded graph is a Delaunay graph, i. e. , whether it is topologically equivalent to a Delaunay triangulation. The problem of recognizing the Delaunay graph had long been open. Recently Hodgson et al. gave a combinatorial characterization of the Delaunay graph, and thus constructed the polynomial-time algorithm for recognizing the Delaunay graphs. Their proof is based on sophisticated discussions on hyperbolic geometry. On the other hand, this paper gives another and simpler proof based on primitive arguments on Euclidean geometry. Moreover, the algorithm is applied to study the distribution of non-Delaunay graphs.