# Bicentric quadrilateral

In Euclidean geometry, a **bicentric quadrilateral** is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called the *inradius* and *circumradius*, and *incenter* and *circumcenter* respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are **chord-tangent quadrilateral**^{[1]} and **inscribed and circumscribed quadrilateral**. It has also been called a **double circle quadrilateral**.^{[2]}

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.^{[3]} This was proved by the French mathematician Jean-Victor Poncelet (1788–1867).

## Special cases

Examples of bicentric quadrilaterals are squares, right kites, and isosceles tangential trapezoids.

## Characterizations

A convex quadrilateral *ABCD* with sides *a*, *b*, *c*, *d* is bicentric if and only if 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 a tangential quadrilateral is tangent to the sides. If the incircle is tangent to the sides *AB*, *BC*, *CD*, *DA* at *W*, *X*, *Y*, *Z* respectively, then a tangential quadrilateral *ABCD* is also cyclic if and only if any one of the following three conditions holds:^{[4]}

*WY*is perpendicular to*XZ*

The first of these three means that the *contact quadrilateral* *WXYZ* is an orthodiagonal quadrilateral.

If *E*, *F*, *G*, *H* are the midpoints of *WX*, *XY*, *YZ*, *ZW* respectively, then the tangential quadrilateral *ABCD* is also cyclic if and only if the quadrilateral *EFGH* is a rectangle.^{[4]}

According to another characterization, if *I* is the incenter in a tangential quadrilateral where the extensions of opposite sides intersect at *J* and *K*, then the quadrilateral is also cyclic if and only if *JIK* is a right angle.^{[4]}

Yet another necessary and sufficient condition is that a tangential quadrilateral *ABCD* is cyclic if and only if its Newton line is perpendicular to the Newton line of its contact quadrilateral *WXYZ*. (The Newton line of a quadrilateral is the line defined by the midpoints of its diagonals.)^{[4]}

## Construction

There is a simple method for constructing a bicentric quadrilateral. Draw two perpendicular chords in a circle (it will be the incircle). At the endpoints of the chords, draw the tangents to the circle. These intersect at four points, which are the vertices of a bicentric quadrilateral.^{[5]} The validity of this construction is due to the characterization that, in a tangential quadrilateral *ABCD*, the contact quadrilateral *WXYZ* has perpendicular diagonals if and only if the tangential quadrilateral is also cyclic.

## Area

### Formulas in terms of four quantities

The area *K* of a bicentric quadrilateral can be expressed in terms of four quantities of the quadrilateral in several different ways. If the sides are *a*, *b*, *c*, *d*, then the area is given by^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}

This is a special case of Brahmagupta's formula. It can also be derived directly from the trigonometric formula for the area of a tangential quadrilateral. Note that the converse does not hold: Some quadrilaterals that are not bicentric also have area ^{[11]} One example of such a quadrilateral is a non-square rectangle.

The area can also be expressed in terms of the tangent lengths *e*, *f*, *g*, *h* as^{[7]}^{:p.128}

A formula for the area of bicentric quadrilateral *ABCD* with incenter *I* is^{[8]}

If a bicentric quadrilateral has tangency chords *k*, *l* and diagonals *p*, *q*, then it has the area^{[7]}^{:p.129}

If *k*, *l* are the tangency chords and *m*, *n* are the bimedians of the quadrilateral, then the area can be calculated using the formula^{[8]}

This formula cannot be used if the quadrilateral is a right kite, since the denominator is zero in that case.

If *M* and *N* are the midpoints of the diagonals, and *E* and *F* are the intersection points of the extensions of opposite sides, then the area of a bicentric quadrilateral is given by

where *I* is the center of the incircle.^{[8]}

### Formulas in terms of three quantities

The area of a bicentric quadrilateral can be expressed in terms of two opposite sides and the angle *θ* between the diagonals according to^{[8]}

In terms of two adjacent angles and the radius *r* of the incircle, the area is given by^{[8]}

The area is given in terms of the circumradius *R* and the inradius *r* as

where *θ* is either angle between the diagonals.^{[12]}

If *M* and *N* are the midpoints of the diagonals, and *E* and *F* are the intersection points of the extensions of opposite sides, then the area can also be expressed as

where *Q* is the foot of the normal to the line *EF* through the center of the incircle.^{[8]}

### Inequalities

If *r* and *R* are the inradius and the circumradius respectively, then the area *K* satisfies the inequalities^{[13]}

There is equality on either side only if the quadrilateral is a square.

Another inequality for the area is^{[14]}^{:p.39,#1203}

where *r* and *R* are the inradius and the circumradius respectively.

A similar inequality giving a sharper upper bound for the area than the previous one is^{[12]}

with equality holding if and only if the quadrilateral is a right kite.

In addition, with sides *a, b, c, d* and semiperimeter *s*:

## Angle formulas

If *a*, *b*, *c*, *d* are the length of the sides *AB*, *BC*, *CD*, *DA* respectively in a bicentric quadrilateral *ABCD*, then its vertex angles can be calculated with the tangent function:^{[8]}

Using the same notations, for the sine and cosine functions the following formulas holds:^{[15]}

The angle *θ* between the diagonals can be calculated from^{[9]}

## Inradius and circumradius

The inradius *r* of a bicentric quadrilateral is determined by the sides *a*, *b*, *c*, *d* according to^{[6]}

The circumradius *R* is given as a special case of Parameshvara's formula. It is^{[6]}

The inradius can also be expressed in terms of the consecutive tangent lengths *e*, *f*, *g*, *h* according to^{[16]}^{:p. 41}

These two formulas are in fact necessary and sufficient conditions for a tangential quadrilateral with inradius *r* to be cyclic.

The four sides *a*, *b*, *c*, *d* of a bicentric quadrilateral are the four solutions of the quartic equation

where *s* is the semiperimeter, and *r* and *R* are the inradius and circumradius respectively.^{[17]}^{:p. 754}

If there is a bicentric quadrilateral with inradius *r* whose tangent lengths are *e*, *f*, *g*, *h*, then there exists a bicentric quadrilateral with inradius *r ^{v}* whose tangent lengths are

*e*,

^{v}*f*,

^{v}*g*,

^{v}*h*, where

^{v}*v*may be any real number.

^{[18]}

^{:pp.9-10}

A bicentric quadrilateral has a greater inradius than does any other tangential quadrilateral having the same sequence of side lengths.^{[19]}^{:pp.392-393}

### Inequalities

The circumradius *R* and the inradius *r* satisfy the inequality

which was proved by L. Fejes Tóth in 1948.^{[18]} It holds with equality only when the two circles are concentric (have the same center as each other); then the quadrilateral is a square. The inequality can be proved in several different ways, one is using the double inequality for the area above.

An extension of the previous inequality is^{[2]}

where there is equality on either side if and only if the quadrilateral is a square.^{[15]}^{:p. 81}

The semiperimeter *s* of a bicentric quadrilateral satisfies^{[18]}^{:p.13}

where *r* and *R* are the inradius and circumradius respectively.

Moreover,^{[14]}^{:p.39,#1203}

and

## Distance between the incenter and circumcenter

### Fuss' theorem

Fuss' theorem gives a relation between the inradius *r*, the circumradius *R* and the distance *x* between the incenter *I* and the circumcenter *O*, for any bicentric quadrilateral. The relation is^{[1]}^{[10]}^{[20]}

or equivalently

It was derived by Nicolaus Fuss (1755–1826) in 1792. Solving for *x* yields

Fuss's theorem 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 *R* and *r* and distance *x* between their centers satisfying the condition in Fuss' theorem, there exists a convex quadrilateral inscribed in one of them and tangent to the other.^{[21]}

Applying to the expression of Fuss's theorem for *x* in terms of *r* and *R* is another way to obtain the above-mentioned inequality A generalization is^{[18]}^{:p.5}

### Carlitz' identity

Another formula for the distance *x* between the centers of the incircle and the circumcircle is due to the American mathematician Leonard Carlitz (1907–1999). It states that^{[22]}

where *r* and *R* are the inradius and the circumradius respectively, and

where *a*, *b*, *c*, *d* are the sides of the bicentric quadrilateral. Carlitz' identity is a generalization of Euler's theorem in geometry to a bicentric quadrilateral.

### Inequalities for the tangent lengths and sides

For the tangent lengths *e*, *f*, *g*, *h* the following inequalities holds:^{[18]}^{:p.3}

and

where *r* is the inradius, *R* is the circumradius, and *x* is the distance between the incenter and circumcenter. The sides *a*, *b*, *c*, *d* satisfy the inequalities^{[18]}^{:p.5}

and

## Other properties of the incenter

The circumcenter, the incenter, and the intersection of the diagonals in a bicentric quadrilateral are collinear.^{[23]}

There is the following equality relating the four distances between the incenter *I* and the vertices of a bicentric quadrilateral *ABCD*:^{[24]}

where *r* is the inradius.

An inequality concerning the inradius *r* and circumradius *R* in a bicentric quadrilateral *ABCD* is^{[25]}

where *I* 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 and a tangential quadrilateral respectively.

In a bicentric quadrilateral with diagonals *p* and *q*, the following identity holds:^{[10]}

where *r* and *R* are the inradius and the circumradius respectively. This equality can be rewritten as^{[12]}

or, solving it as a quadratic equation for the product of the diagonals, in the form

An inequality for the product of the diagonals *p*, *q* in a bicentric quadrilateral is^{[13]}

where *a*, *b*, *c*, *d* are the sides. This was proved by Murray S. Klamkin in 1967.

## See also

## References

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