Bicentric quadrilateral

In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. This means they have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names are chord-tangent quadrilateral^[1] and inscribed and circumscribed quadrilateral.

1 Special cases
2 Characterizations
3 Area
4 Angle formulas
5 Inradius and circumradius
6 Fuss' theorem and Carlitz' identity
7 Other properties
8 See also
9 References

Special cases

Examples of bicentric quadrilaterals are squares, right kites and those isosceles trapezoids where the length of the legs is the arithmetic mean of the bases.

Characterizations

A convex quadrilateral ABCD with sides a, b, c, d is bicentric if and only if opposite sides satisfy Pitot's theorem and opposite angles are supplementary, that is $\displaystyle a%2Bc=b%2Bd$ and $\displaystyle A%2BC=B%2BD=\pi.$

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 of^[2]

$WY\bot\, XZ$ (That is, the contact quadrilateral WXYZ is an orthodiagonal quadrilateral.)

$\frac{AW}{WB}=\frac{DY}{YC}$

$\frac{AC}{BD}=\frac{AW%2BCY}{BX%2BDZ}$

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.^[2]

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.^[2]

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.)^[2]

Area

The area K of a bicentric quadrilateral with sides a, b, c, d is^[3] ^[4] ^[5] ^[6] ^[7]

$\displaystyle K = \sqrt{abcd}.$

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.

If a bicentric quadrilateral has tangency chords k, l and diagonals p, q, then it has the area^[4]

$K=\frac{klpq}{k^2%2Bl^2}.$

Another formula for the area is^[5]

$K=\left|\frac{m^2-n^2}{k^2-l^2}\right|kl$

where m and n are the bimedians of the quadrilateral.

The area can also be expressed in terms of the tangent lengths e, f, g, h as^[4]

$K=\sqrt[4]{efgh}(e%2Bf%2Bg%2Bh).$

Yet another formula for the area of bicentric quadrilateral ABCD is^[5]

$K=AI\cdot CI%2BBI\cdot DI$

where I is the center of the incircle. In terms of two adjacent angles and the radius r of the incircle, the area is given as^[5]

$K=2r^2\left(\frac{1}{\sin{A}}%2B\frac{1}{\sin{B}}\right).$

If r and R are the inradius and the circumradius respectively, then the area K satisfies the inequalities^[8]

$\displaystyle 4r^2 \le K \le 2R^2.$

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

Another inequality for the area is

$K \le \tfrac{4}{3}r\sqrt{4R^2%2Br^2}$

where r and R are the inradius and the circumradius respectively.^[9]

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 are given by^[5]

$\tan{\frac{A}{2}}=\sqrt{\frac{bc}{ad}}=\cot{\frac{C}{2}},$

$\tan{\frac{B}{2}}=\sqrt{\frac{cd}{ab}}=\cot{\frac{D}{2}}.$

The angle $\theta$ between the diagonals can be calculated from^[6]

$\displaystyle \tan{\frac{\theta}{2}}=\sqrt{\frac{bd}{ac}}.$

Inradius and circumradius

The inradius r of a bicentric quadrilateral is determined by the sides a, b, c, d according to^[3]

$\displaystyle r=\frac{\sqrt{abcd}}{a%2Bc}.$

The inradius can also be expressed in terms of the consecutive tangent lengths e, f, g, h according to^[10]^{:p. 41}

$\displaystyle r=\sqrt{eg}=\sqrt{fh}.$

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

The circumradius R is given as a special case of Parameshvara's formula. It is^[3]

$\displaystyle R=\frac{1}{4}\sqrt{\frac{(ab%2Bcd)(ac%2Bbd)(ad%2Bbc)}{abcd}}.$

The two radii satisfy the inequality $R\ge \sqrt{2}r$ . It holds with equality only when the two circles are concentric (have the same center as each other); then the quadrilateral is a square. This inequality follows from the double inequality for the area.

Another inequality concerning the two radii in a bicentric quadrilateral ABCD is

$4r^2 \le IA\cdot IC%2BIB\cdot ID \le 2R^2$

where I is the incenter.^[11]

In a bicentric quadrilateral with diagonals p and q, it holds that^[7]

$\displaystyle \frac{pq}{4r^2}-\frac{4R^2}{pq}=1$

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

Fuss' theorem and Carlitz' identity

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] ^[7] ^[12]

$\frac{1}{(R-x)^2}%2B\frac{1}{(R%2Bx)^2}=\frac{1}{r^2},$

or equivalently

$\displaystyle 2r^2(R^2%2Bx^2)=(R^2-x^2)^2$ .

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

$x=\sqrt{R^2%2Br^2-r\sqrt{4R^2%2Br^2}}.$

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.^[13]

Applying $x^2 \ge 0$ to the expression of Fuss's theorem for x in terms of r and R is another way to obtain the above-mentioned inequality $R \ge \sqrt{2}r.$

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^[14]

$\displaystyle x^2=R^2-2Rr\cdot \mu$

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

$\displaystyle \mu=\sqrt{\frac{(ab%2Bcd)(ad%2Bbc)}{(a%2Bc)^2(ac%2Bbd)}} = \sqrt{\frac{(ab%2Bcd)(ad%2Bbc)}{(b%2Bd)^2(ac%2Bbd)}}$

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.

Other properties

The circumcenter, the incenter, and the intersection of the diagonals in a bicentric quadrilateral are collinear.^[15]

An inequality for the product of the diagonals p, q in a bicentric quadrilateral is

$\displaystyle 8pq\le (a%2Bb%2Bc%2Bd)^2$

where a, b, c, d are the sides. This was proved by Murray Klamkin in 1967.^[8]

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.^[16] This was proved by the French mathematician Jean-Victor Poncelet (1788–1867).

The semiperimeter s satisfies^[9]

$2\sqrt{K} \le s \le r %2B \sqrt{4R^2%2Br^2}$

where K is the area of the bicentric quadrilateral and r, R are the inradius and circumradius respectively.

References

^ ^a ^b Dörrie, Heinrich, 100 Great Problems of Elementary Mathematics: Their History and Solutions, New York: Dover, 1965, pp. 188–193.
^ ^a ^b ^c ^d Josefsson, Martin (2010), "Characterizations of Bicentric Quadrilaterals", Forum Geometricorum 10: 165–173, http://forumgeom.fau.edu/FG2010volume10/FG201019.pdf .
^ ^a ^b ^c Weisstein, Eric, Bicentric Quadrilateral at MathWorld, [1], Accessed on 2011-08-13.
^ ^a ^b ^c Josefsson, Martin (2010), "Calculations concerning the tangent lengths and tangency chords of a tangential quadrilateral", Forum Geometricorum 10: 119–130, http://forumgeom.fau.edu/FG2010volume10/FG201013.pdf .
^ ^a ^b ^c ^d ^e Josefsson, Martin (2011), "The Area of a Bicentric Quadrilateral", Forum Geometricorum 11: 155–164, http://forumgeom.fau.edu/FG2011volume11/FG201116.pdf .
^ ^a ^b Durell, C. V. and Robson, A., Advanced Trigonometry, Dover, 2003, pp. 28, 30.
^ ^a ^b ^c Yiu, Paul, Euclidean Geometry, [2], 1998, pp. 158-164.
^ ^a ^b Alsina, Claudi and Nelsen, Roger, When less is more: visualizing basic inequalities, Mathematical Association of America, 2009, pp. 64-66.
^ ^a ^b Inequalities proposed in “Crux Mathematicorum”, 2007, Problem 1203, p. 39, [3]
^ M. Radic, Z. Kaliman, and V. Kadum, "A condition that a tangential quadrilateral is also a chordal one", Mathematical Communications, 12 (2007) 33–52.
^ Post at Art of Problem Solving, 2009, [4]
^ Salazar, Juan Carlos (2006), "Fuss's Theorem", Mathematical Gazette 90 (July): 306–307 .
^ Byerly, W. E. (1909), "The In- and-Circumscribed Quadrilateral", The Annals of Mathematics 10: 123–128 .
^ Calin, Ovidiu, Euclidean and Non-Euclidean Geometry a metric approach, [5], pp. 153–158.
^ Bogomolny, Alex, Collinearity in Bicentric Quadrilaterals [6], 2004.
^ Weisstein, Eric W. "Poncelet Transverse." From MathWorld--A Wolfram Web Resource, [7]