In the previous chapter of this book, a beneficial and practical topic related to the triangulation of a set of points into Delaunay triangles was discussed for a planar set of points. In this chapter, for the planar set of points \({\mathcal{P}} = P_{1} ,P_{2} , \cdots ,P_{n}\) , very efficient algorithms are considered, which, among other things, will answer the question of which point in the planar set \({\mathcal{P}}\) is closest to an arbitrarily chosen point P outside that set. For example, suppose the points in the set \({\mathcal{P}}\) represent specific centers (e.g., schools, clinics, etc.), and point P represents a location (GPS coordinate) of a pedestrian. In that case, the algorithms discussed in this chapter, related to searching Voronoi diagrams, will quickly answer questions like, ``Which clinic or school is closest to the pedestrian P?” This concept pertains to the proximity problem, precisely the nearest neighbor problem, which involves the wealthy geometry of the Voronoi diagram. It is important to note that many natural and social phenomena can be described using Voronoi diagrams, such as forestry, crystallography, ecology, art, geophysics, meteorology, geodesy, economics, biophysics, and more. In 1644, Descartes utilized Voronoi diagrams, while 1850 Dirichlet used them to study quadratic forms. In 1854, British physician John Snow used a Voronoi diagram to demonstrate that most people who died in a cholera epidemic lived closer to the infected Broad Street pump than any other water pump. The Voronoi diagram was also studied by Russian mathematician Georgy Fedosseevich Voronoi, who 1908 defined and examined generalized n-dimensional spaces. More information about the applications of Voronoi diagrams can be found at the link: https://www.voronoi.com.au . Moreover, there is a profound connection between the Voronoi diagram and Delaunay triangulation and their relationship with the 3D convex hull, which will be discussed later in this chapter.