An equilateral triangle
A bicentric kite
A bicentric isosceles trapezoid
A regular pentagon
In geometry, a bicentric polygon is a tangential
with unequal sides is not bicentric, because no circle can be tangent to all four sides.
Triangles
Every triangle is bicentric.
1
R
−
x
+
1
R
+
x
=
1
r
{\displaystyle {\frac {1}{R-x}}+{\frac {1}{R+x}}={\frac {1}{r}}}
where x is the distance between the centers of the circles.[2] 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
{\displaystyle R>r}
, there exists a convex quadrilateral inscribed in one of them and tangent to the other if and only if their radii satisfy
1
(
R
−
x
)
2
+
1
(
R
+
x
)
2
=
1
r
2
{\displaystyle {\frac {1}{(R-x)^{2}}}+{\frac {1}{(R+x)^{2}}}={\frac {1}{r^{2}}}}
where x is the distance between their centers.[2] [3] This condition (and analogous conditions for higher order polygons) is known as Fuss' theorem .[4]
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.[5] Some of these for specific n are:
n
=
5
:
r
(
R
−
x
)
=
(
R
+
x
)
(
R
−
r
+
x
)
(
R
−
r
−
x
)
+
(
R
+
x
)
2
R
(
R
−
r
−
x
)
,
{\displaystyle n=5:\quad r(R-x)=(R+x){\sqrt {(R-r+x)(R-r-x)}}+(R+x){\sqrt {2R(R-r-x)}},}
n
=
6
:
3
(
R
2
−
x
2
)
4
=
4
r
2
(
R
2
+
x
2
)
(
R
2
−
x
2
)
2
+
16
r
4
x
2
R
2
,
{\displaystyle n=6:\quad 3(R^{2}-x^{2})^{4}=4r^{2}(R^{2}+x^{2})(R^{2}-x^{2})^{2}+16r^{4}x^{2}R^{2},}
n
=
8
:
16
p
4
q
4
(
p
2
−
1
)
(
q
2
−
1
)
=
(
p
2
+
q
2
−
p
2
q
2
)
4
,
{\displaystyle n=8:\quad 16p^{4}q^{4}(p^{2}-1)(q^{2}-1)=(p^{2}+q^{2}-p^{2}q^{2})^{4},}
where
p
=
R
+
x
r
{\displaystyle p={\tfrac {R+x}{r}}}
and
q
=
R
−
x
r
.
{\displaystyle q={\tfrac {R-x}{r}}.}
Regular polygons
Every
concentric—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
are:
R
=
a
2
sin
π
n
=
r
cos
π
n
.
{\displaystyle R={\frac {a}{2\sin {\frac {\pi }{n}}}}={\frac {r}{\cos {\frac {\pi }{n}}}}.}
For some regular polygons which can be
for these relations:
n
{\displaystyle n\!\,}
R
and
a
{\displaystyle R\,{\text{and}}\,a\!\,}
r
and
a
{\displaystyle r\,{\text{and}}\,a\!\,}
r
and
R
{\displaystyle r\,{\text{and}}\,R\!\,}
3
R
3
=
a
{\displaystyle R{\sqrt {3}}=a\!\,}
2
r
=
a
3
3
{\displaystyle 2r={\frac {a}{3}}{\sqrt {3}}\!\,}
2
r
=
R
{\displaystyle 2r=R\!\,}
4
R
2
=
a
{\displaystyle R{\sqrt {2}}=a\!\,}
r
=
a
2
{\displaystyle r={\frac {a}{2}}\!\,}
2
r
=
R
2
{\displaystyle 2r=R{\sqrt {2}}\!\,}
5
R
5
−
5
2
=
a
{\displaystyle R{\sqrt {\frac {5-{\sqrt {5}}}{2}}}=a\!\,}
r
(
5
−
1
)
=
a
10
50
+
10
5
{\displaystyle r\left({\sqrt {5}}-1\right)={\frac {a}{10}}{\sqrt {50+10{\sqrt {5}}}}\!\,}
r
(
5
−
1
)
=
R
{\displaystyle r({\sqrt {5}}-1)=R\!\,}
6
R
=
a
{\displaystyle R=a\!\,}
2
r
3
3
=
a
{\displaystyle {\frac {2r}{3}}{\sqrt {3}}=a\!\,}
2
r
3
3
=
R
{\displaystyle {\frac {2r}{3}}{\sqrt {3}}=R\!\,}
8
R
2
+
2
=
a
(
2
+
1
)
{\displaystyle R{\sqrt {2+{\sqrt {2}}}}=a\left({\sqrt {2}}+1\right)\!\,}
r
4
−
2
2
=
a
2
4
+
2
2
{\displaystyle r{\sqrt {4-2{\sqrt {2}}}}={\frac {a}{2}}{\sqrt {4+2{\sqrt {2}}}}\!\,}
2
r
(
2
−
1
)
=
R
2
−
2
{\displaystyle 2r\left({\sqrt {2}}-1\right)=R{\sqrt {2-{\sqrt {2}}}}\!\,}
10
(
5
−
1
)
R
=
2
a
{\displaystyle ({\sqrt {5}}-1)R=2a\!\,}
2
r
25
−
10
5
=
5
a
{\displaystyle 2r{\sqrt {25-10{\sqrt {5}}}}=5a\!\,}
2
r
5
25
−
10
5
=
R
2
(
5
−
1
)
{\displaystyle {\frac {2r}{5}}{\sqrt {25-10{\sqrt {5}}}}={\frac {R}{2}}\left({\sqrt {5}}-1\right)\!\,}
Thus we have the following decimal approximations:
n
{\displaystyle n\!\,}
R
/
a
{\displaystyle R/a\!\,}
r
/
a
{\displaystyle r/a\!\,}
R
/
r
{\displaystyle R/r\!\,}
3
{\displaystyle 3\,}
0.577
{\displaystyle 0.577\,}
0.289
{\displaystyle 0.289}
2.000
{\displaystyle 2.000\,}
4
{\displaystyle 4}
0.707
{\displaystyle 0.707\,}
0.500
{\displaystyle 0.500}
1.414
{\displaystyle 1.414\,}
5
{\displaystyle 5}
0.851
{\displaystyle 0.851\,}
0.688
{\displaystyle 0.688}
1.236
{\displaystyle 1.236\,}
6
{\displaystyle 6}
1.000
{\displaystyle 1.000\,}
0.866
{\displaystyle 0.866}
1.155
{\displaystyle 1.155\,}
8
{\displaystyle 8}
1.307
{\displaystyle 1.307\,}
1.207
{\displaystyle 1.207}
1.082
{\displaystyle 1.082\,}
10
{\displaystyle 10}
1.618
{\displaystyle 1.618\,}
1.539
{\displaystyle 1.539}
1.051
{\displaystyle 1.051\,}
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
Moreover, given a circumcircle and incircle, each diagonal of the variable polygon is tangent to a fixed circle. [7]
References
.
^ .
^ Davison, Charles (1915), Subjects for mathematical essays , Macmillan and co., limited, p. 98 .
.
^ Weisstein, Eric W. "Poncelet's Porism." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PonceletsPorism.html
.
^ Johnson, Roger A. Advanced Euclidean Geometry , Dover Publ., 2007 (1929), p. 94.
External links