![Bicentric quadrilateral poncelet](https://upload.wikimedia.org/wikipedia/commons/2/25/Bicentric_quadrilateral_poncelet.svg)
In
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, a bicentric quadrilateral is a
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
that has both an
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. ...
and a
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 ...
. The radii and center of these circles are called ''inradius'' and ''circumradius'', and ''incenter'' and ''circumcenter'' respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both
tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the ...
s and
cyclic quadrilateral
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
s. Other names for these quadrilaterals are chord-tangent quadrilateral
and inscribed and circumscribed quadrilateral. It has also rarely been called a ''double circle quadrilateral''
and ''double scribed quadrilateral''.
If two circles, one within the other, are the incircle and the circumcircle of a bicentric quadrilateral, then every point on the circumcircle is the vertex of a bicentric quadrilateral having the same incircle and circumcircle. This is a special case of
Poncelet's porism
In geometry, Poncelet's closure theorem, also known as Poncelet's porism, states that whenever a polygon is inscribed figure, inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygo ...
, which was proved by the French mathematician
Jean-Victor Poncelet
Jean-Victor Poncelet (; 1 July 1788 – 22 December 1867) was a French engineer and mathematician who served most notably as the Commanding General of the École Polytechnique. He is considered a reviver of projective geometry, and his work ''Tr ...
(1788–1867).
Special cases
![Bicentric kite 001](https://upload.wikimedia.org/wikipedia/commons/3/37/Bicentric_kite_001.svg)
Examples of bicentric quadrilaterals are
squares
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 ...
,
right kite
In Euclidean geometry, a right kite is a kite (a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other) that can be inscribed in a circle.Michael de Villiers, ''Some Adventures in Eucl ...
s, and
isosceles tangential trapezoids.
Characterizations
![Bicentric quadrilateral](https://upload.wikimedia.org/wikipedia/commons/e/ef/Bicentric_quadrilateral.svg)
A convex quadrilateral with sides is bicentric
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
opposite sides satisfy
Pitot's theorem for tangential quadrilaterals ''and'' the cyclic quadrilateral property that opposite angles are
supplementary; that is,
:
Three other characterizations concern the points where 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. ...
in a
tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the ...
is tangent to the sides. If the incircle is tangent to the sides at respectively, then a tangential quadrilateral is also cyclic if and only if any one of the following three conditions holds:
[.]
* is
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to
*
*
The first of these three means that the ''contact quadrilateral'' is an
orthodiagonal quadrilateral
In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular ...
.
If are the midpoints of respectively, then the tangential quadrilateral is also cyclic
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
the quadrilateral is a
rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...
.
According to another characterization, if is the
incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
in a
tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the ...
where the extensions of opposite sides intersect at and , then the quadrilateral is also cyclic if and only if is a
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
.
Yet another
necessary and sufficient condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
is that a tangential quadrilateral is cyclic if and only if its
Newton line
In Euclidean geometry the Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral with at most two parallel sides.Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. ...
is perpendicular to the Newton line of its contact quadrilateral . (The Newton line of a quadrilateral is the line defined by the midpoints of its diagonals.)
Construction
![01-Bicentric quadrilateral](https://upload.wikimedia.org/wikipedia/commons/0/05/01-Bicentric_quadrilateral.svg)
]
There is a simple method for constructing a bicentric quadrilateral:
It starts with the incircle around the
Centre (geometry), centre with the radius and then draw two to each other
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
chords
Chord may refer to:
* Chord (music), an aggregate of musical pitches sounded simultaneously
** Guitar chord a chord played on a guitar, which has a particular tuning
* Chord (geometry), a line segment joining two points on a curve
* Chord ( ...
and in the incircle . At the endpoints of the chords draw the
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is 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 the curve. More ...
s to the incircle. These intersect at four points , which are the
vertices of a bicentric quadrilateral.
To draw the circumcircle, draw two
perpendicular bisector
In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
s on the sides of the bicentric quadrilateral respectively . The perpendicular bisectors intersect in the centre of the circumcircle with the distance to the centre of the incircle . The circumcircle can be drawn around the centre .
The validity of this construction is due to the characterization that, in a
tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the ...
, the contact quadrilateral has perpendicular
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
s if and only if the tangential quadrilateral 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 s ...
.
Area
Formulas in terms of four quantities
The
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
of a bicentric quadrilateral can be expressed in terms of four quantities of the quadrilateral in several different ways. If the sides are , then the area is given by
:
This is a special case of
Brahmagupta's formula
In Euclidean geometry, Brahmagupta's formula is used to find the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides; its generalized version (Bretschneider's formula) can be used with non-cyclic ...
. It can also be derived directly from the trigonometric formula for the area of a
tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the ...
. Note that the converse does not hold: Some quadrilaterals that are not bicentric also have area
One example of such a quadrilateral is a non-square
rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...
.
The area can also be expressed in terms of the
tangent lengths as
:
A formula for the area of bicentric quadrilateral with incenter is
:
If a bicentric quadrilateral has
tangency chords and diagonals , then it has area
[.]
:
If are the tangency chords and are the
bimedians of the quadrilateral, then the area can be calculated using the formula
[.]
:
This formula cannot be used if the quadrilateral is a
right kite
In Euclidean geometry, a right kite is a kite (a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other) that can be inscribed in a circle.Michael de Villiers, ''Some Adventures in Eucl ...
, since the denominator is zero in that case.
If are the midpoints of the diagonals, and are the intersection points of the extensions of opposite sides, then the area of a bicentric quadrilateral is given by
:
where is the center of the incircle.
Formulas in terms of three quantities
The area of a bicentric quadrilateral can be expressed in terms of two opposite sides and the angle between the diagonals according to
:
In terms of two adjacent angles and the radius of the incircle, the area is given by
:
The area is given in terms of the circumradius and the inradius as
:
where is either angle between the diagonals.
[.]
If are the midpoints of the diagonals, and are the intersection points of the extensions of opposite sides, then the area can also be expressed as
:
where is the foot of the perpendicular to the line through the center of the incircle.
Inequalities
If and are the inradius and the circumradius respectively, then the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
satisfies the
inequalities
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
:
There is equality on either side only if the quadrilateral is a
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 adj ...
.
Another inequality for the area is
[Inequalities proposed in '']Crux Mathematicorum
''Crux Mathematicorum'' is a scientific journal of mathematics published by the Canadian Mathematical Society. It contains mathematical problems for secondary school and undergraduate students. , its editor-in-chief is Kseniya Garaschuk.
The journ ...
'', 200
/ref>
:
where and are the inradius and the circumradius respectively.
A similar inequality giving a sharper upper bound for the area than the previous one is
:
with equality holding if and only if the quadrilateral is a right kite
In Euclidean geometry, a right kite is a kite (a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other) that can be inscribed in a circle.Michael de Villiers, ''Some Adventures in Eucl ...
.
In addition, with sides and semiperimeter
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...
:
:
:
:
Angle formulas
If are the length of the sides respectively in a bicentric quadrilateral , then its vertex angles can be calculated with the tangent function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...
:
:
Using the same notations, for the sine and cosine functions the following formulas holds:[.]
:
The angle between the diagonals can be calculated from[Durell, C. V. and Robson, A., ''Advanced Trigonometry'', Dover, 2003, pp. 28, 30.]
:
Inradius and circumradius
The inradius
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 a bicentric quadrilateral is determined by the sides according to[Weisstein, Eric, Bicentric Quadrilateral at ''MathWorld'']
Accessed on 2011-08-13.
:
The circumradius
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 ...
is given as a special case of Parameshvara
Vatasseri Parameshvara Nambudiri ( 1380–1460) was a major Indian mathematician and astronomer of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama. He was also an astrologer. Parameshvara was a proponent of ob ...
's formula. It is
:
The inradius can also be expressed in terms of the consecutive tangent lengths according to
:
These two formulas are in fact necessary and sufficient condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
s for a tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the ...
with inradius to be 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 s ...
.
The four sides of a bicentric quadrilateral are the four solutions of the quartic equation
In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is
:ax^4+bx^3+cx^2+dx+e=0 \,
where ''a'' ≠ 0.
The quartic is the highest order polynomi ...
:
where is the semiperimeter, and and are the inradius and circumradius respectively.[Pop, Ovidiu T., "Identities and inequalities in a quadrilateral", ''Octogon Mathematical Magazine'', Vol. 17, No. 2, October 2009, pp 754-763.]
If there is a bicentric quadrilateral with inradius whose tangent lengths are , then there exists a bicentric quadrilateral with inradius whose tangent lengths are where may be any real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
.
A bicentric quadrilateral has a greater inradius than does any other tangential quadrilateral having the same sequence of side lengths.[.]
Inequalities
The circumradius and the inradius satisfy the inequality
:
which was proved by L. Fejes Tóth in 1948.[Radic, Mirko, "Certain inequalities concerning bicentric quadrilaterals, hexagons and octagons", ''Journal of Inequalities in Pure and Applied Mathematics'', Volume 6, Issue 1, 2005]
/ref> It holds with equality only when the two circles 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 point ...
(have the same center as each other); then the quadrilateral is a 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 adj ...
. The inequality can be proved in several different ways, one using the double inequality for the area above.
An extension of the previous inequality is[Yun, Zhang, "Euler's Inequality Revisited", ''Mathematical Spectrum'', Volume 40, Number 3 (May 2008), pp. 119-121. First page available a]
.
:
where there is equality on either side if and only if the quadrilateral is a 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 adj ...
.
The semiperimeter
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...
of a bicentric quadrilateral satisfies
:
where and are the inradius and circumradius respectively.
Moreover,
:
and
:
Distance between the incenter and circumcenter
Fuss' theorem
Fuss' theorem gives a relation between the inradius
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.
...
, the circumradius
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 ...
and the distance between the incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
and the circumcenter
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 ...
, for any bicentric quadrilateral. The relation is[Yiu, Paul, ''Euclidean Geometry'']
1998, pp. 158-164.
:
or equivalently
:
It was derived by Nicolaus Fuss (1755–1826) in 1792. Solving for yields
:
Fuss's theorem, which is the analog of Euler's theorem for triangles for bicentric quadrilaterals, says that if a quadrilateral is bicentric, then its two associated circles are related according to the above equations. In fact the converse also holds: given two circles (one within the other) with radii and and distance between their centers satisfying the condition in Fuss' theorem, there exists a convex quadrilateral inscribed in one of them and tangent to the other (and then 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 i ...
, there exist infinitely many of them).
Applying to the expression of Fuss's theorem for in terms of and is another way to obtain the above-mentioned inequality A generalization is
:
Carlitz' identity
Another formula for the distance between the centers of 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. ...
and 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 ...
is due to the American mathematician Leonard Carlitz
Leonard Carlitz (December 26, 1907 – September 17, 1999) was an American mathematician. Carlitz supervised 44 doctorates at Duke University and published over 770 papers.
Chronology
* 1907 Born Philadelphia, PA, USA
* 1927 BA, University ...
(1907–1999). It states that
:
where and are the inradius
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.
...
and the circumradius
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, and
:
where are the sides of the bicentric quadrilateral.
Inequalities for the tangent lengths and sides
For the tangent lengths the following inequalities holds:
:
and
:
where is the inradius, is the circumradius, and is the distance between the incenter and circumcenter. The sides satisfy the inequalities
:
and
:
Other properties of the incenter
The circumcenter
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 ...
, the incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
, and the intersection of the diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
s in a bicentric quadrilateral 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 ...
.
There is the following equality relating the four distances between the incenter and the vertices of a bicentric quadrilateral :
:
where is the inradius.
If is the intersection of the diagonals in a bicentric quadrilateral with incenter , then
:
An inequality concerning the inradius and circumradius in a bicentric quadrilateral is
:
where is the incenter.
Properties of the diagonals
The lengths of the diagonals in a bicentric quadrilateral can be expressed in terms of the sides or the tangent lengths, which are formulas that holds in a cyclic quadrilateral
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
and a tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the ...
respectively.
In a bicentric quadrilateral with diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
s , the following identity holds:
:
where and are the inradius
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.
...
and the circumradius
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. This equality can be rewritten as
:
or, solving it as a quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
for the product of the diagonals, in the form
:
An inequality for the product of the diagonals in a bicentric quadrilateral is
:
where are the sides. This was proved by Murray S. Klamkin in 1967.
Four incenters lie on a circle
Let be a bicentric quadrilateral and the center of its circumcircle. Then the incenters of the four triangles lie on a circle.[Alexey A. Zaslavsky, One property of bicentral quadrilaterals, 2019]
/ref>
See also
*Bicentric polygon
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 — that is, inscribed in an outer circle that passes through each vertex of the polygon. All triang ...
*Ex-tangential quadrilateral
In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the ''extensions'' of all four sides are tangent to a circle outside the quadrilateral.Radic, Mirko; Kaliman, Zoran and Kadum, Vladimir, "A condition that a tan ...
References
{{Polygons
Types of quadrilaterals