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Since $$\\widetilde{P}$$<\/jats:tex-math>\n \n P<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is infinite, the algorithms to compute Delaunay triangulations in the plane do not generalize naturally. Using a Dirichlet domain, we exhibit a finite set of points that captures the full triangulation. We prove that an edge of a Delaunay triangulation has a combinatorial length (a notion we define in the paper) smaller than $$12g-6$$<\/jats:tex-math>\n \n 12<\/mml:mn>\n g<\/mml:mi>\n -<\/mml:mo>\n 6<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with respect to a Dirichlet domain. To achieve this, we introduce new tools, of intrinsic interest, that capture the properties of length-minimizing curves in the context of closed curves. We then use these to derive structural results on Delaunay triangulations and exhibit certain distance minimizing properties of both the edges of a Delaunay triangulation and of a Dirichlet domain. The bounds produced in this paper depend only on the topology of the surface. 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