Bicentric quadrilateral

Bicentric quadrilateral
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers 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 quadrilaterals and cyclic quadrilaterals. 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, which 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 with sides 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 at respectively, then a tangential quadrilateral is also cyclic if and only if any one of the following three conditions holds:
is perpendicular to
The first of these three means that the contact quadrilateral is an orthodiagonal quadrilateral.
If are the midpoints of respectively, then the tangential quadrilateral is also cyclic if and only if the quadrilateral is a rectangle.
According to another characterization, if is the incenter in a tangential quadrilateral where the extensions of opposite sides intersect at and , then the quadrilateral is also cyclic if and only if is a right angle.
Yet another necessary and sufficient condition is that a tangential quadrilateral is cyclic if and only if its Newton line 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
There is a simple method for constructing a bicentric quadrilateral:
It starts with the incircle around the centre with the radius and then draw two to each other perpendicular chords and in the incircle . At the endpoints of the chords draw the tangents to the incircle. These intersect at four points , which are the vertices of a bicentric quadrilateral.
To draw the circumcircle, draw two perpendicular bisectors 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 , the contact quadrilateral has perpendicular diagonals if and only if the tangential quadrilateral is also cyclic.
Area
Formulas in terms of four quantities
The area 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. 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 One example of such a quadrilateral is a non-square rectangle.
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, 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 satisfies the inequalities
There is equality on either side only if the quadrilateral is a square.
Another inequality for the area is
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 addition, with sides and semiperimeter :
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:
Using the same notations, for the sine and cosine functions the following formulas holds:
The angle between the diagonals can be calculated from
Inradius and circumradius
The inradius of a bicentric quadrilateral is determined by the sides according to
The circumradius is given as a special case of Parameshvara'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 conditions for a tangential quadrilateral with inradius to be cyclic.
The four sides of a bicentric quadrilateral are the four solutions of the quartic equation
where is the semiperimeter, and and are the inradius and circumradius respectively.
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.
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. 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 using the double inequality for the area above.
An extension of the previous inequality is
where there is equality on either side if and only if the quadrilateral is a square.
The semiperimeter 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 , the circumradius and the distance between the incenter and the circumcenter , for any bicentric quadrilateral. The relation is
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, 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 and the circumcircle is due to the American mathematician Leonard Carlitz (1907–1999). It states that
where and are the inradius and the circumradius 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, the incenter, and the intersection of the diagonals in a bicentric quadrilateral are collinear.
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 and a tangential quadrilateral respectively.
In a bicentric quadrilateral with diagonals , the following identity holds:
where and are the inradius and the circumradius respectively. This equality can be rewritten as
or, solving it as a quadratic equation 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.
See also
Bicentric polygon
Ex-tangential quadrilateral
References
Types of quadrilaterals