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 polygon edges and faces in orthogonal projections "m"-cubes. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. The list gives the number of solutions for smaller polygons. The area "A" of a convex regular "n"-sided polygon having side "s", circumradius "R", apothem "a", and perimeter "p" is given by For regular polygons with side "s" = 1, circumradius "R" = 1, or apothem "a" = 1, this produces the following table: (Note that since formula_11 as formula_12, the area when formula_13 is tending to formula_14 as formula_15 grows large.) Of