In geometry, a bicentric polygon is a
tangential
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on ...
polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain.
The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
(a polygon all of whose sides are tangent to an inner
incircle) which is also
cyclic — that is,
inscribed in an
outer circle that passes through each vertex of the polygon. All
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 ...
s and all
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 ...
s are bicentric. On the other hand, a
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 ...
with unequal sides is not bicentric, because no circle can be tangent to all four sides.
Triangles
Every triangle is bicentric. In a triangle, the radii ''r'' and ''R'' of the
incircle and
circumcircle
In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...
respectively are related by the
equation
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...
:
where ''x'' is the distance between the centers of the circles.
[.] This is one version of
Euler's triangle formula.
Bicentric quadrilaterals
Not all
quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
s are bicentric (having both an incircle and a circumcircle). Given two circles (one within the other) with radii ''R'' and ''r'' where
, there exists a convex quadrilateral inscribed in one of them and tangent to the other
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
their radii satisfy
:
where ''x'' is the distance between their centers.
This condition (and analogous conditions for higher order polygons) is known as
Fuss' theorem.
Polygons with n > 4
A complicated general formula is known for any number ''n'' of sides for the relation among the circumradius ''R'', the inradius ''r'', and the distance ''x'' between the circumcenter and the incenter. Some of these for specific ''n'' are:
:
:
:
where
and
Regular polygons
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 bicentric.
In a regular polygon, the incircle and the circumcircle are
concentric
In geometry, two or more objects are said to be ''concentric'' when they share the same center. Any pair of (possibly unalike) objects with well-defined centers can be concentric, including circles, spheres, regular polygons, regular polyh ...
—that is, they share a common center, which is also the center of the regular polygon, so the distance between the incenter and circumcenter is always zero. The radius of the inscribed circle is the
apothem (the shortest distance from the center to the boundary of the regular polygon).
For any regular polygon, the relations between the common
edge length ''a'', the radius ''r'' of the
incircle, and the radius ''R'' of the
circumcircle
In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...
are:
:
For some regular polygons which can be
constructed with compass and ruler, we have the following
algebraic formulas for these relations:
Thus we have the following decimal approximations:
Poncelet's porism
If two circles are the inscribed and circumscribed circles of a particular bicentric ''n''-gon, then the same two circles are the inscribed and circumscribed circles of infinitely many bicentric ''n''-gons. More precisely,
every
tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to the inner of the two circles can be extended to a bicentric ''n''-gon by placing vertices on the line at the points where it crosses the outer circle, continuing from each vertex along another tangent line, and continuing in the same way until the resulting
polygonal chain
In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...
closes up to an ''n''-gon. The fact that it will always do so is implied by
Poncelet's closure theorem, which more generally applies for inscribed and circumscribed
conics.
Moreover, given a circumcircle and incircle, each diagonal of the variable polygon is tangent to a fixed circle.
[Johnson, Roger A. ''Advanced Euclidean Geometry'', Dover Publ., 2007 (1929), p. 94.]
References
External links
* {{MathWorld, title=Bicentric polygon, urlname=BicentricPolygon
Elementary geometry
Types of polygons