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In geometry, a bicentric polygon is a tangential polygon (a polygon all of whose sides are tangent to an inner incircle) which is also
cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in so ...
— that is,
inscribe {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...
d in an outer circle that passes through each vertex of the polygon. All triangles and all regular polygons are bicentric. On the other hand, a rectangle 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 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 polyg ...
respectively are related by the equation :\frac+\frac=\frac where ''x'' is the distance between the centers of the circles.. This is one version of Euler's triangle formula.


Bicentric quadrilaterals

Not all quadrilaterals are bicentric (having both an incircle and a circumcircle). Given two circles (one within the other) with radii ''R'' and ''r'' where R>r, there exists a convex quadrilateral inscribed in one of them and tangent to the other if and only if their radii satisfy :\frac+\frac=\frac 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: :n=5: \quad r(R-x)=(R+x)\sqrt+(R+x)\sqrt , :n=6: \quad 3(R^2-x^2)^4=4r^2(R^2+x^2)(R^2-x^2)^2+16r^4x^2R^2 , :n=8: \quad 16p^4q^4(p^2-1)(q^2-1)=(p^2+q^2-p^2q^2)^4 , where p=\tfrac and q=\tfrac.


Regular polygons

Every regular polygon is bicentric. In a regular polygon, the incircle and the circumcircle are
concentric In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center poin ...
—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 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 polyg ...
are: :R=\frac=\frac. 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 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 closes up to an ''n''-gon. The fact that it will always do so is implied by
Poncelet's closure theorem In geometry, Poncelet's closure theorem, also known as Poncelet's porism, states that whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygons that are all in ...
, which more generally applies for inscribed and circumscribed
conic In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
s. 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