Applying Delaunay Triangulation in Olympiad Problems

Some fundamental properties of Delaunay triangulation are presented in this article. The construction of the triangulation described here has not been encountered by the authors in the known literature. The main point is to apply this triangulation in solving otherwise challenging Olympiad problems in combinatorial geometry. One of the problems in which the method is applied is from the ’Mikl´os Schweitzer’ competition, 2002. It is extremely difficult without knowing the Delaunay triangulation properties. The second problem illustrating the method is from a prestigious Russian 239 competition held in 2024. The third is from the USA team selection tests for the IMO 2024.