In mathematics and particularly in

elementary geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

, a circumgon is a geometric figure which circumscribes some

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

, in the sense that it is the union of the outer edges of non-overlapping triangles each of which has a vertex at the center of the circle and opposite side on a line that is tangent to the circle. The limiting case in which part or all of the circumgon is a

circular arc Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fallacy * Circular ...

is permitted. A circumgonal region is the union of those triangular regions. Every

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

is a circumgonal region because it circumscribes the circle known as the

incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

of the triangle. Every

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

is a circumgonal region. In fact, every

regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

is a circumgonal region, as is more generally every

tangential polygon In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an ''incircle''). This is a circle that is tangent to each of the polygon's sides. The dual pol ...

. But not every polygon is a circumgonal region: for example, a non-square rectangle is not. A circumgonal region need not even be a

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...

: for example, it could consist of three triangular wedges meeting only at the circle's center. All circumgons have common properties regarding area–perimeter ratios and centroids. It is these properties that make circumgons interesting objects of study in elementary geometry. The concept and the terminology of a circumgon were introduced and their properties investigated first by Tom M. Apostol and Mamikon A. Mnatsakanian in a paper published in 2004.

Properties

Given a circumgon, the circle which the circumgon circumscribes is called the ''incircle'' of the circumgon, the radius of the circle is called the ''inradius'', and its center is called the ''incenter''. *The area of a circumgonal region is equal to half the product of its perimeter (the total length of the outer edges) and its inradius. *The vector from the incenter to the area centroid, ''G_A'' , of a circumgonal region and the vector from the incenter to the centroid of its boundary (outer edge points), ''G_B'' , are related by ::

G_B = \tfracG_A.

:Thus the two centroids and the incenter are

collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...

References

{{reflist Circles Triangle geometry Geometric shapes