Cyclic Quadrilateral

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1. Cyclic Quadrilateral By - Archit Agarwal IX-A 11

2. Introduction to 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   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 .

3. Properties of a Cyclic Quadrilateral The opposite angles of a cyclic quadrilateral are supplementary. or The sum of either pair of opposite angles of a cyclic quadrilateral is 180 If one side of a cyclic quadrilateral are produced, then the exterior angle will be equal to the opposite interior angle. If the sum of any pair of opposite angles of a quadrilateral is 180 , then the quadrilateral is cyclic.  

4. Area of Cyclic Quadrilateral The area K of a cyclic quadrilateral with sides a, b, c, d is given by Brahmagupta's formula. k=√ (s-a) (s-b) (s-c) (s-d) Where s, the semi perimeter, is = ½ ( a+b+c+d ) . 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.

5. 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.

6. 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 = ¼ √(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=√{(ab+cd)(ac+bd)(ad+bc)}

7. Theorems of Cyclic Quadrilateral 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. A B D A C B D C

8. Proving the Cyclic Quadrilateral Theorem- Part 1 The opposite angles of a cyclic quadrilateral are supplementary. A B C D Inscribed Angle Inscribed Angle Sum of Arcs Prove that

9. Proving the Cyclic Quadrilateral Theorem- Part 2 An exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. (Opposite angles of a cyclic Quadrilateral) (Supplementary Angle Theorem) Prove that 1 2 3 4 5

10. Using the Cyclic Quadrilateral Theorem 82 103 1 2 3 1. _______ 2. _______ 3. _______ 41 49 28

11. Using the Cyclic Quadrilateral Theorem 100 35 1 2 3 4 5 6 7 8 9 80 80 35 35 110 100 100 30 30

12. 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.