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In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and particularly in elementary geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a circumgon is a geometric figure which circumscribes some circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
, 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
A circular arc is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than radians (180 ...
is permitted. A circumgonal region is the union of those triangular regions.
Every triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
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 geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
is a circumgonal region. In fact, every regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
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 po ...
. But not every polygon is a circumgonal region: for example, a non-square rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
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 ...
: 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, ''GA'' , of a circumgonal region and the vector from the incenter to the centroid of its boundary (outer edge points), ''GB'' , are related by :: :Thus the two centroids and the incenter arecollinear
In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
.
See Also
*Concyclic points
In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. A polygon whose vertices are concyclic is called a cyclic polygon, and the circle is called its ''circumscribing circle'' or ''circumcircle'' ...
References
{{reflist Circles Triangle geometry Geometric shapes