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 skew polyhedron Regular skew polyhedron In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces. Infinite regular skew polyhedra that span 3-space or higher are called regular skew apeirohedra. According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to "regular skew polyhedra". Coxeter offered a modified Schläfli symbol {l,m|n} for these figures, with {l,m} implying the vertex figure,