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From: To: Hitesh Kumar Prabhakhar Sir Durga Prasad Mathematics Department IX ‘B’ JSS Publ ic School J .S .S publ ic school , Bage Bage

Introduction To Cyclic Quadrilaterals

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called thecircumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. 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".All triangles have a circumcircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be cyclic is a non-square rhombus.

Properties of a Cyclic Quadrilateral

1. The opposite angles of a cyclic quadrilateral are supplementary.

orThe sum of either pair of opposite angles of a cyclic quadrilateral is 1800

2. If one side of a cyclic quadrilateral are produced, then the exterior angle will be equal to the opposite interior angle.

3. If the sum of any pair of opposite angles of a quadrilateral is 1800, then the quadrilateral is cyclic.

Area of a Cyclic Quadrilateral

The area K of a cyclic quadrilateral with sides a, b, c, d is given by Brahmagupta's formula.

Where s, the semi perimeter, is . 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.

Four unequal lengths, each less than the sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals, 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.

Parameshvara's Formula

A cyclic quadrilateral with successive sides a, b, c, d and semi perimeter s has the circumradius (the radius of the circumcircle) given by

R=\frac{1}{4} \sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}.This was derived by the Indian mathematician Vatasseri Parameshvara in the 15th century.

Using Brahmagupta's formula, Parameshvara's formula can be restated as

4KR=\sqrt{(ab+cd)(ac+bd)(ad+bc)}

Cyclic Quadrilateral Theorem

The opposite angles of a cyclic quadrilateral are supplementary. An exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

Theorems of Cyclic Quadrilateral

B

D

A D1800

C B1800A

C

B

D

BDECABx

x

The opposite angles of a cyclic quadrilateral are supplementary.

A

B

CD

Prove that

BC 1800.

ABD ACD3600

C

1

2ABD

InscribedAngle

B

1

2ACD Inscribed

Angle

Sum ofArcs

1

2ABD

1

2ACD1800

Thus, B + C =1800.

Proving the Cyclic Quadrilateral Theorem- Part 1

An exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

1

2

3

45

2 4 1800

Opposite angles of a cyclicquadrilateral

4 5 1800

Supplementary Angle Theorem

4 5 2 4

Thus, 5 = 2.Prove that

2 = 5.

Transitive Property

Proving the Cyclic Quadrilateral Theorem- Part 2

820

1030

1

23

1. _______

2. _______

3. _______

410

490

280

Using the Cyclic Quadrilateral Theorem

1000

350

1

2

3

4

5

6

7

1. _______

2. _______

3. _______

4. _______

5. _______

6. _______

7. _______

8. _______

9. _______

800

800

1000

1000

350

350

1100

8

9300

300

Using the Cyclic Quadrilateral Theorem

Conclusion

Finally we conclude that this given PPT on Cyclic Quadrilaterals was very helpful, educational, and was fun too. So we thank our mathematics teacher for giving us this PPT assignment. While creating this PPT we had a great time while doing it and while sharing our ideas.