Cyclic quadrilateral

Cyclic quadrilaterals.

In Euclidean geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. Other names for these quadrilaterals are chordal quadrilateral and inscribed quadrilateral. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

The word cyclic is from the Greek "kuklos" which means circle or wheel.

Special cases

Any square, rectangle, isosceles trapezoid, or antiparallelogram is cyclic. A kite is cyclic if and only if it has two right angles. A bicentric quadrilateral is a cyclic quadrilateral that is also tangential and an ex-bicentric quadrilateral is a cyclic quadrilateral that is also ex-tangential.

Characterizations

A convex quadrilateral is cyclic if and only if the four perpendicular bisectors to the sides are concurrent. This common point is the circumcenter.

A convex quadrilateral ABCD is cyclic if and only if its opposite angles are supplementary, that is $A + C = B + D = \pi = 180^{\circ}$.^[1] Equivalently, a convex quadrilateral is cyclic if and only if each exterior angle is equal to the opposite interior angle.

Another necessary and sufficient condition for a convex quadrilateral ABCD to be cyclic is that an angle between a side and a diagonal is equal to the angle between the opposite side and the other diagonal.^[2] That is, for example, $\angle ACB = \angle ADB.$

Yet another characterization is that a convex quadrilateral ABCD is cyclic if and only if^[3]

$\tan{\frac{A}{2}}\tan{\frac{C}{2}}=\tan{\frac{B}{2}}\tan{\frac{D}{2}}.$

Area

The area K of a cyclic quadrilateral with sides a, b, c, d is given by Brahmagupta's formula^[4]:p.24

$K=\sqrt{(s-a)(s-b)(s-c)(s-d)} \,$

where s, the semiperimeter, is $s=\frac{a+b+c+d}{2}$. It is a corollary to Bretschneider's formula since opposite angles are supplementary. If also d = 0, the cyclic quadrilateral becomes a triangle and the formula is reduced to Heron's formula.

The cyclic quadrilateral has maximal area among all quadrilaterals having the same sequence of side lengths. This is another corollary to Bretschneider's formula. It can also be proved using calculus.^[5]

Four unequal lengths, each less than then sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals,^[6] which by Brahmagupta's formula all have the same area. Specifically, for sides a, b, c, and d, side a could be opposite any of side b, side c, or side d.

The area of a cyclic quadrilateral with successive sides a, b, c, d and angle B between sides a and b can be expressed as^[4]:p.25

$K = \tfrac{1}{2}(ab+cd)\sin{B}$

or^[4]:p.26

$K = \tfrac{1}{2}(ac+bd)\sin{\theta}$

where θ is the angle between the diagonals. If A is an oblique angle, the area can also be expressed as^[4]:p.26

$K = \tfrac{1}{4}(a^2-b^2-c^2+d^2)\tan{A}.$

Another formula is^[7]:p.83

$\displaystyle K=2R^2\sin{A}\sin{B}\sin{\theta}$

where R is the radius in the circumcircle.

In terms of the sides a, b, c, d, the area satisfies the inequality^[8]

$\displaystyle K\le \tfrac{1}{2}\sqrt[3]{(ab+cd)(ac+bd)(ad+bc)}.$

Diagonals

Ptolemy's theorem expresses the product of the lengths of the two diagonals p and q of a cyclic quadrilateral as equal to the sum of the products ac and bd of opposite sides:^[4]:p.25

$\displaystyle pq = ac + bd.$

The converse is also true. That is, if this equation is satisfied in a convex quadrilateral, then it is a cyclic quadrilateral. Thus Ptolemy's theorem is another characterization of cyclic quadrilaterals.

In any convex quadrilateral, the two diagonals together partition the quadrilateral into four triangles; in a cyclic quadrilateral, opposite pairs of these four triangles are similar to each other.

Ptolemy's second theorem states that a cyclic quadrilateral with successive vertices A, B, C, D and sides a = AB, b = BC, c = CD, and d = DA, and with diagonals p = AC and q = BD, has^[4]:p.25

$\frac {p}{q}= \frac{ad+cb}{ab+cd}.$

The lengths of the diagonals are given in terms of the sides (using the same notations as above) as^[4]:p.25

$p = \sqrt{\frac{(ac+bd)(ad+bc)}{ab+cd}},$

and

$q = \sqrt{\frac{(ac+bd)(ab+dc)}{ad+bc}}.$

For the sum of the diagonals we have the inequality^[9]

$p+q\ge 2\sqrt{ac+bd}.$

Equality holds if and only if the diagonals have equal length, which can be proved using the AM-GM inequality.

If the intersection of the diagonals divides one diagonal into segments of lengths e and f, and divides the other diagonal into segments of lengths g and h, then ef = gh. This is known as the intersecting chords theorem.

Angle formulas

For a cyclic quadrilateral with successive sides a, b, c, d, semiperimeter s, and angle A between sides a and d, the trigonometric functions of A are given by^[10]

$\cos A = \frac{a^2 + d^2 - b^2 - c^2}{2(ad + bc)},$

$\sin A = \frac{2\sqrt{(s-a)(s-b)(s-c)(s-d)}}{(ad+bc)},$

$\tan \frac{A}{2} = \sqrt{\frac{(s-a)(s-d)}{(s-b)(s-c)}}.$

The angle θ between the diagonals satisfies^[4]:p.26

$\tan \frac{\theta}{2} = \sqrt{\frac{(s-b)(s-d)}{(s-a)(s-c)}}.$

If the extensions of opposite sides a and c intersect at an angle φ, then

$\cos{\frac{\varphi}{2}}=\sqrt{\frac{(s-b)(s-d)(b+d)^2}{(ab+cd)(ad+bc)}}$

where s is the semiperimeter.^[4]:p.31

Parameshvara's formula

A cyclic quadrilateral with successive sides a, b, c, d and semiperimeter s has circumradius (the radius of the circumcircle) given by^[11]

$R=\frac{1}{4} \sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}.$

It was derived by the Indian mathematician Vatasseri Parameshvara in the fifteenth century.

Other properties

• There are no cyclic quadrilaterals with rational area and with unequal rational sides in either arithmetic or geometric progression.^[12]

• Four line segments, each perpendicular to one side of a cyclic quadrilateral and passing through the opposite side's midpoint, are concurrent.^[13]:p.131 These line segments are called the maltitudes,^[14] which is an abbreviation for midpoint altitude.

• If the opposite sides of a cyclic quadrilateral are extended to meet at E and F, then the internal angle bisectors of the angles at E and F are perpendicular.^[6] If M and N are the midpoints of the diagonals AC and BD, then^[15]

$\frac{MN}{EF}=\frac{1}{2}\left |\frac{AC}{BD}-\frac{BD}{AC}\right|.$

• In a cyclic quadrilateral ABCD, the incenters in triangles ABC, BCD, CDA, and DAB are the vertices of a rectangle, see Japanese theorem for cyclic quadrilaterals.

Properties of cyclic quadrilaterals that are also orthodiagonal

• Brahmagupta's theorem states that for a cyclic quadrilateral that is also orthodiagonal (has mutually perpendicular diagonals), the perpendicular from any side through the point of intersection of the diagonals bisects the other side.^[13]:p.137

• If a cyclic quadrilateral is also orthodiagonal, the distance from the circumcenter (the center of the circumscribed circle) to any side equals half the length of the opposite side.^[13]:p.138

• For a cyclic orthodiagonal quadrilateral, suppose the intersection of the diagonals divides one diagonal into segments of lengths p₁ and p₂ and divides the other diagonal into segments of lengths q₁ and q₂. Then^[16]

$p_1^2+p_2^2+q_1^2+q_2^2=a^2+c^2=b^2+d^2=D^2$

where D is the diameter of the circumcircle. This holds because the diagonals are perpendicular chords of a circle. Equivalently, letting R = D / 2 be the radius of the circumcircle, the average of $p_1^2,$ $p_2^2,$ $q_1^2,$ and $q_2^2$ is R². Moreover, the equations a² + c² = b² + d² = D² imply that in an orthodiagonal cyclic quadrilateral, the sum of the squares of the sides equals eight times the square of the circumradius.

• If an orthodiagonal quadrilateral is also cyclic, then the midpoints of the sides and the feet of the perpendiculars from these midpoints to the opposite sides lie on a circle centered at the centroid of the quadrilateral. This circle is called the eight point circle.

See also

• Bicentric quadrilateral
• Brahmagupta's theorem on perpendicular diagonals of cyclic quadrilaterals
• Ex-tangential quadrilateral
• Cyclic polygon
• Japanese theorem for cyclic quadrilaterals
• Orthodiagonal quadrilateral
• Ptolemy's table of chords
• Ptolemy's theorem
• Tangential quadrilateral, a quadrilateral all of whose sides are tangent to a single circle

References

1. ^ Book 3, Proposition 22 of Euclid's Elements.
2. ^ Andreescu, Titu and Enescu, Bogdan, Mathematical Olympiad Treasures, Birkhäuser, 2006, p. 44.
3. ^ Hajja, Mowaffaq (2008), "A Condition for a Circumscriptible Quadrilateral to be Cyclic", Forum Geometricorum 8: 103–106, http://forumgeom.fau.edu/FG2008volume8/FG200814.pdf .
4. ^ ^{a b c d e f g h i} Durell, C. V. and Robson, A., Advanced trigonometry, Dover, 2003.
5. ^ Thomas Peter, Maximizing the Area of a Quadrilateral, The College Mathematics Journal, Vol. 34, No. 4 (Sep., 2003), pp. 315-316.
6. ^ ^{a b} Coxeter, H. S. M. and Greitzer, S. L., Geometry Revisited, 1967, pp. 57, 60.
7. ^ Viktor Prasolov, Problems in plane and solid geometry: v.1 Plane Geometry, [1]
8. ^ Alsina, Claudi and Nelsen, Roger, When less is more: visualizing basic inequalities, Mathematical Association of America, 2009, p. 66.
9. ^ Inequalities proposed in “Crux Mathematicorum”, 2007, Problem 2975, p. 123, [2]
10. ^ Siddons, A. W., and R. T. Hughes, Trigonometry, Cambridge Univ. Press, 1929: p. 202.
11. ^ Hoehn, Larry, "Circumradius of a cyclic quadrilateral," Mathematical Gazette 84, March 2000, 69–70.
12. ^ Buchholz, R. H., and MacDougall, J. A. "Heron quadrilaterals with sides in arithmetic or geometric progression", Bull. Austral. Math. Soc. 59 (1999), 263–269. http://journals.cambridge.org/article_S0004972700032883
13. ^ ^{a b c} Altshiller-Court, College Geometry, Dover Publ., 2007
14. ^ Weisstein, Eric, Maltitude, Mathworld, [3]. Accessed 2011-08-16.
15. ^ Post at Art of Problem Solving, 2010, [4]
16. ^ Posamentier, Alfred S., and Charles T. Salkind, Challenging Problems in Geometry, Dover Publ., second edition, 1996:pp. 104–105, #4–23.

External links

• Derivation of Formula for the Area of Cyclic Quadrilateral
• Incenters in Cyclic Quadrilateral at cut-the-knot
• Four Concurrent Lines in a Cyclic Quadrilateral at cut-the-knot
• Weisstein, Eric W., "Cyclic quadrilateral" from MathWorld.