Click a triangle or an edge and the reading panel tells you what it shows: the empty circle property, or the dual pairing. Right-click to place a point, drag to move it, and shift-click to remove it. The wheel changes how large the diagram is shown, and dragging with the middle button, or with space held, moves it around. The stage panel switches between convex hull, Delaunay, and Voronoi, and each stage stays visible under the next one.
Pick a triangle to see its circumcircle. No other point ever falls inside it, which is the empty circle property that defines a Delaunay triangulation. Pick an edge instead and the Voronoi edge that crosses it at a right angle appears: the two diagrams are the same information seen from opposite sides.
Among all possible triangulations, Delaunay is the one that makes the thinnest triangle as fat as it can be, so it maximises the smallest angle. The shortest tree joining the points, the minimum spanning tree, is also built only from Delaunay edges.
Several of the point layouts are degenerate on purpose. On a circle, a square lattice, or concentric rings, points share one circle, so the circumcircle test comes out as a tie. On the square lattice there is then no way to choose which diagonal splits each square, and jittering the same lattice slightly resolves the tie and fixes the orientation. On a line no triangle forms at all and the cells become strips, while lifting a single point off that line gives one fan of triangles. Relaxing moves each point toward the centroid of its own cell, and repeating it pushes the cells toward hexagons.