Incenter centers known to the ancient Greeks, and the only one that does not in general lie on the Euler line. It is the first listed center, X(1), in Clark Kimberling's Encyclopedia of Triangle Centers, and the identity element of the multiplicative group of triangle centers. For polygons with more than three sides, the incenter only exists for tangential polygons—those that have an incircle that is tangent to each side of the polygon. In this case the incenter is the center of this circle and is equally distant from all sides. It is a theorem in Euclidean geometry that the three
Regular 4-polytope are 4 dual-pairs and 2 self-dual forms among the ten regular star polychora. Note: The cells (polyhedra), their faces (polygons), the "polygonal edge figures" and "polyhedral vertex figures" are identified by their Schläfli symbols. Regular 4-polytope In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogs of the regular polyhedra in three dimensions and the regular polygons in two dimensions. Regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century, although the full set were not discovered until later. There are six convex and ten star regular 4-polytopes, giving a