Bicentric quadrilateral
Convex, 4-sided shape with an incircle and a circumcircle
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 [1] and inscribed and circumscribed quadrilateral . It has also rarely been called a double circle quadrilateral [2] and double scribed quadrilateral . [3]
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. [4] 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 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: [5]
- 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 . [5]
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 . [5]
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.) [5]
Construction
There is a simple method for constructing a bicentric quadrilateral:
It starts with the incircle C r around the centre I with the radius r and then draw two to each other perpendicular chords WY and XZ in the incircle C r . At the endpoints of the chords draw the tangents a, b, c, d to the incircle. These intersect at four points A, B, C, D , which are the vertices of a bicentric quadrilateral. [6] To draw the circumcircle, draw two perpendicular bisectors p 1 , p 2 on the sides of the bicentric quadrilateral a respectively b . The perpendicular bisectors p 1 , p 2 intersect in the centre O of the circumcircle C R with the distance x to the centre I of the incircle C r . The circumcircle can be drawn around the centre O .
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 [7] [8] [9] [10] [11]
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 [12] 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 [8] : p.128
A formula for the area of bicentric quadrilateral ABCD with incenter I is [9]
If a bicentric quadrilateral has tangency chords k, l and diagonals p, q , then it has area [8] : 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 [9]
This formula cannot be used if the quadrilateral is a right kite , since the denominator is zero in that case.
If M, N are the midpoints of the diagonals, and E, 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. [9]
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 [9]
In terms of two adjacent angles and the radius r of the incircle, the area is given by [9]
The area is given in terms of the circumradius R and the inradius r as
where θ is either angle between the diagonals. [13]
If M, N are the midpoints of the diagonals, and E, 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 perpendicular to the line EF through the center of the incircle. [9]
Inequalities
If r and R are the inradius and the circumradius respectively, then the area K satisfies the inequalities [14]
There is equality on either side only if the quadrilateral is a square .
Another inequality for the area is [15] : 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 [13]
with equality holding if and only if the quadrilateral is a right kite .
In addition, with sides a, b, c, d and semiperimeter s :
- [15] : p.39, #1203
- [15] : p.39, #1203
- [15] : p.39, #1203
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 : [9]
Using the same notations, for the sine and cosine functions the following formulas holds: [16]
The angle θ between the diagonals can be calculated from [10]
Inradius and circumradius
The inradius r of a bicentric quadrilateral is determined by the sides a, b, c, d according to [7]
The circumradius R is given as a special case of Parameshvara 's formula. It is [7]
The inradius can also be expressed in terms of the consecutive tangent lengths e, f, g, h according to [17] : 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. [18] : 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 where v may be any real number . [19] : pp.9–10
A bicentric quadrilateral has a greater inradius than does any other tangential quadrilateral having the same sequence of side lengths. [20] : pp.392–393
Inequalities
The circumradius R and the inradius r satisfy the inequality
which was proved by L. Fejes Tóth in 1948. [19] 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 [2] [21] : p. 141
where there is equality on either side if and only if the quadrilateral is a square . [16] : p. 81
The semiperimeter s of a bicentric quadrilateral satisfies [19] : p.13
where r and R are the inradius and circumradius respectively.
Moreover, [15] : p.39, #1203
and
- [15] : p.62, #1599
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] [11] [22]
or equivalently
It was derived by Nicolaus Fuss (1755–1826) in 1792. Solving for x 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 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 [23] (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 [19] : 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 [24]
where r and R are the inradius and the circumradius respectively, and
where a, b, c, d are the sides of the bicentric quadrilateral.
Inequalities for the tangent lengths and sides
For the tangent lengths e, f, g, h the following inequalities holds: [19] : 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 [19] : p.5
and
Other properties of the incenter
The circumcenter , the incenter , and the intersection of the diagonals in a bicentric quadrilateral are collinear . [25]
There is the following equality relating the four distances between the incenter I and the vertices of a bicentric quadrilateral ABCD : [26]
where r is the inradius.
If P is the intersection of the diagonals in a bicentric quadrilateral ABCD with incenter I , then [27]
An inequality concerning the inradius r and circumradius R in a bicentric quadrilateral ABCD is [28]
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, q , the following identity holds: [11]
where r and R are the inradius and the circumradius respectively. This equality can be rewritten as [13]
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 [14]
where a, b, c, d are the sides. This was proved by Murray S. Klamkin in 1967.
Four incenters lie on a circle
Let ABCD be a bicentric quadrilateral and O the center of its circumcircle. Then the incenters of the four triangles △ OAB , △ OBC , △ OCD , △ ODA lie on a circle. [29]
See also
References
- 1 2 Dörrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics: Their History and Solutions . New York: Dover. pp. 188–193. ISBN 978-0-486-61348-2 .
- 1 2 Yun, Zhang, "Euler's Inequality Revisited", Mathematical Spectrum , Volume 40, Number 3 (May 2008), pp. 119-121. First page available at Archived March 4, 2016, at the Wayback Machine .
- ↑ Leng, Gangsong (2016). Geometric Inequalities: In Mathematical Olympiad and Competitions . Shanghai: East China Normal University Press. p. 22. ISBN 978-981-4704-13-7 .
- ↑ Weisstein, Eric W. "Poncelet Transverse." From MathWorld – A Wolfram Web Resource,
- 1 2 3 4 Josefsson, Martin (2010), "Characterizations of Bicentric Quadrilaterals" (PDF) , Forum Geometricorum , 10 : 165–173 .
- ↑ Alsina, Claudi; Nelsen, Roger (2011). Icons of Mathematics. An exploration of twenty key images . Mathematical Association of America. pp. 125–126. ISBN 978-0-88385-352-8 .
- 1 2 3 Weisstein, Eric, Bicentric Quadrilateral at MathWorld , , Accessed on 2011-08-13.
- 1 2 3 Josefsson, Martin (2010), "Calculations concerning the tangent lengths and tangency chords of a tangential quadrilateral" (PDF) , Forum Geometricorum , 10 : 119–130 .
- 1 2 3 4 5 6 7 8 Josefsson, Martin (2011), "The Area of a Bicentric Quadrilateral" (PDF) , Forum Geometricorum , 11 : 155–164 .
- 1 2 Durell, C. V. and Robson, A., Advanced Trigonometry , Dover, 2003, pp. 28, 30.
- 1 2 3 Yiu, Paul, Euclidean Geometry , , 1998, pp. 158-164.
- ↑ Lord, Nick, "Quadrilaterals with area formula ", Mathematical Gazette 96, July 2012, 345-347.
- 1 2 3 Josefsson, Martin (2012), "Maximal Area of a Bicentric Quadrilateral" (PDF) , Forum Geometricorum , 12 : 237–241 .
- 1 2 Alsina, Claudi; Nelsen, Roger (2009). When less is more: visualizing basic inequalities . Mathematical Association of America. pp. 64 –66. ISBN 978-0-88385-342-9 .
- 1 2 3 4 5 6 Inequalities proposed in Crux Mathematicorum , 2007.
- 1 2 Josefsson, Martin (2012), "A New Proof of Yun's Inequality for Bicentric Quadrilaterals" (PDF) , Forum Geometricorum , 12 : 79–82 .
- ↑ M. Radic, Z. Kaliman, and V. Kadum, "A condition that a tangential quadrilateral is also a chordal one", Mathematical Communications , 12 (2007) 33–52.
- ↑ Pop, Ovidiu T., "Identities and inequalities in a quadrilateral", Octogon Mathematical Magazine , Vol. 17, No. 2, October 2009, pp 754-763.
- 1 2 3 4 5 6 Radic, Mirko, "Certain inequalities concerning bicentric quadrilaterals, hexagons and octagons", Journal of Inequalities in Pure and Applied Mathematics , Volume 6, Issue 1, 2005,
- ↑ Hess, Albrecht (2014), "On a circle containing the incenters of tangential quadrilaterals" (PDF) , Forum Geometricorum , 14 : 389–396 .
- ↑ Shattuck, Mark, “A Geometric Inequality for Cyclic Quadrilaterals”, Forum Geometricorum 18, 2018, 141-154. This paper also gives various inequalities in terms of the arc lengths subtended by a cyclic quadrilateral’s sides.
- ↑ 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, doi : 10.2307/1967103 .
- ↑ Calin, Ovidiu, Euclidean and Non-Euclidean Geometry a metric approach , , pp. 153–158.
- ↑ Bogomolny, Alex, Collinearity in Bicentric Quadrilaterals , 2004.
- ↑ L. V. Nagarajan, Bi-centric Polygons , 2014, .
- ↑ Crux Mathematicorum 34 (2008) no 4, p. 242.
- ↑ Post at Art of Problem Solving , 2009
- ↑ Alexey A. Zaslavsky, One property of bicentral quadrilaterals, 2019,
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