In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polygon has a circumscribed circle. A

**polygon that**does have one is called a cyclic polygon, or sometimes a concyclic polygon because its**vertices**are concyclic. All triangles, all regular simple polygons, all rectangles, all isosceles trapezoids, and all right- Triangles
All triangles are

**cyclic**; that is, every triangle has a... **Cyclic**quadrilateralsQuadrilaterals that can be circumscribed have particular...

**Cyclic**n-gonsFor a

**cyclic polygon**with an odd number of sides, all angles...

- Triangles
Japanese theorem for cyclic polygons From Wikipedia, the free encyclopedia In geometry, the Japanese theorem states that

**no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant. sum of the radii of the green**circles**= sum of the radii of the red**circles- Overview
- Etymology
- Classification
- Properties and formulas
- Generalizations

In geometry, a polygon is

**a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit.**The solid plane region, the bounding circuit, or the two together, may be called a polygon. The segments of a polygonal circuit are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners. The interior of a solid polygon is sometimes called its body. An n-gon is a polygon with n sides;The word polygon derives from the Greek adjective πολύς 'much', 'many' and γωνία 'corner' or 'angle'. It has been suggested that γόνυ 'knee' may be the origin of gon.

Polygons are primarily classified by the number of sides. See the table below.

Polygons may be characterized by their convexity or type of non-convexity: 1. Convex: any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the ...

Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are: 1. Interior angle – The sum of the interior angles of a simple n-gon is π radians or × 180 degrees. This is because any simple n-gon can be considered to be ...

In this section, the vertices of the polygon under consideration are taken to be,, …, {\\displaystyle,,\\ldots,} in order. For convenience in some formulas, the notation = will also be used. If the polygon is non-self-intersecting, the signed area is A = 1 2 ∑ i = 0 n ...

Using the same convention for vertex coordinates as in the previous section, the coordinates of the centroid of a solid simple polygon are 1. C x = 1 6 A ∑ i = 0 n − 1, {\\displaystyle C_{x}={\\frac {1}{6A}}\\sum _{i=0}^{n-1},} 2. C y = 1 6 A ∑ i = 0 n − 1 ...

The idea of a

**polygon**has been generalized in various ways. Some of the more important include: 1. A spherical**polygon**is a circuit of arcs of great circles and vertices on the surface of a sphere. It allows the digon, a**polygon**having only two sides and two corners, which is impossible in a flat plane. Spherical polygons play an important role in cartography and in Wythoff's construction of the uniform polyhedra. 2. A skew**polygon**does not lie in a flat plane, but zigzags in three dimensions. T**en.wikipedia.org**A polygon is defined to be cyclic if

**its vertices are all concyclic.**For example, all the vertices of a regular polygon of any number of sides are concyclic. A tangential polygon is one having an inscribed circle tangent to each side of the polygon; these tangency points are thus concyclic on the inscribed circle.Special cases. Any square, rectangle, isosceles trapezoid, or antiparallelogram is

**cyclic**. A kite is**cyclic**if and only if it has two right angles. A bicentric quadrilateral is a**cyclic**quadrilateral that is also tangential and an ex-bicentric quadrilateral is a**cyclic**quadrilateral that is also ex-tangential.**A polygon whose vertices are located on a common circle.**McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill... Explanation of cyclic polygonExample equiangular

**polygons**Direct Indirect Skew A rectangle, <4>, is a convex direct equiangular**polygon**, containing four 90° internal angles.: A concave indirect equiangular**polygon**, <6-2>, like this hexagon, counterclockwise, has five left turns and one right turn, like this tetromino.A convex polygon is called concyclic or a cyclic poly

**gon if all the vertices lie on a single circle.**A cyclic and equilateral polygon is called regular; for each number of sides, all regular polygons with the same number of sides are similar. A simple polygon may also be defined as regular if it is cyclic and equilateral.A polygon is defined to be cyclic if

**its vertices are all concyclic.**For example, all the vertices of a regular polygon of any number of sides are concyclic. A tangential polygon is one having an inscribed circle tangent to each side of the polygon; these tangency points are thus concyclic on the inscribed circle.