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 and inscribed and circumscribed quadrilateral. It has also been called a double circle 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 corollary of Poncelet's porism, which was proved by the French mathematician Jean-Victor Poncelet (1788–1867).
- 1 Special cases
- 2 Characterizations
- 3 Construction
- 4 Area
- 5 Angle formulas
- 6 Inradius and circumradius
- 7 Distance between the incenter and circumcenter
- 8 Other properties of the incenter
- 9 Properties of the diagonals
- 10 See also
- 11 References
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:
- WY is perpendicular to XZ
The first of these three means that the contact quadrilateral WXYZ is an orthodiagonal quadrilateral.
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.
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.)
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. 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.
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
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.
A formula for the area of bicentric quadrilateral ABCD with incenter I is
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.
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
In terms of two adjacent angles and the radius r of the incircle, the area is given by
The area is given in terms of the circumradius R and the inradius r as
where θ is either angle between the diagonals.
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.
There is equality on either side only if the quadrilateral is a square.
Another inequality for the area is: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
with equality holding if and only if the quadrilateral is a right kite.
In addition, with sides a, b, c, d and semiperimeter s:
The angle θ between the diagonals can be calculated from
Inradius and circumradius
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.: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 rv whose tangent lengths are ev, fv, gv, hv, where v may be any real number.:pp.9-10
A bicentric quadrilateral has a greater inradius than does any other tangential quadrilateral having the same sequence of side lengths.:pp.392-393
The circumradius R and the inradius r 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 is using the double inequality for the area above.
An extension of the previous inequality is
where r and R are the inradius and circumradius respectively.
Distance between the incenter and circumcenter
It was derived by Nicolaus Fuss (1755–1826) in 1792. Solving for x yields
Fuss's theorem, which is a generalization of Euler's theorem in geometry to a bicentric quadrilateral, 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 (and then by Poncelet's closure theorem, there exist infinitely many of them).
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:p.5
where a, b, c, d are the sides of the bicentric quadrilateral.
Inequalities for the tangent lengths and sides
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:p.5
Other properties of the incenter
There is the following equality relating the four distances between the incenter I and the vertices of a bicentric quadrilateral ABCD:
where r is the inradius.
If P is the intersection of the diagonals in a bicentric quadrilateral ABCD with incenter I, then
An inequality concerning the inradius r and circumradius R in a bicentric quadrilateral ABCD is
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.
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
where a, b, c, d are the sides. This was proved by Murray S. Klamkin in 1967.
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