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
Truncation (geometry) like a pentagon, but is actually a double-covered (degenerate) decagon ({10/2}) with two sets of overlapping vertices and edges. A truncated great heptagram {7/3} gives a tetradecagram {14/3}. When the term applies to truncating platonic solids or regular tilings, usually "uniform truncation" is implied, which means truncating until the original faces become regular polygons with twice as many sides as the original form. This sequence shows an example of the truncation of a cube, using four steps of a continuous truncating process between a full cube and a rectified cube. The final polyhedron is a cuboctahedron. The middle image is