Does the Delaunay triangulation contain all triangles with empty circumcircle?

Question

I have a set of points and I want to find all triangles that have an empty circumcircle. I think that that the Delaunay triangulation does so.

I have read some papers on the subject but I am not sure whether the Delaunay triangulation finds all such triangles. If yes, then how can I mathematically prove that?

Answer 1

Here is a proof: Assume that p, q, r are three points of P not on a common line such that no other point of P is in the circle C defined by p, q, r. Then the center of C is a Voronoi node of the Voronoi diagram V(P) of P. Note that V(P) is the dual graph of the Delaunay triangulation D(P) of P: Every Voronoi node belongs to a Delaunay triangle (and vice versa). The dual of the node mentioned above is your triangle.

See "Computational Geometry" by de Berg, Cheong, van Kreveld, Overmars for basic properties on Voronoi diagrams and Delaunay triangulations.