BodhiAI

Nov. 12, 2019

**Cyclic Quadrilateral**

**A quadrilateral inscribed in a circle is called a cyclic quadrilateral. Sum of opposite angles of cyclic quadrilateral is 1800.**

**A. Relation Between Sides & Angles of Cyclic Quadrilateral**

**The figure represents a cyclic quadrilateral ABCD and a, b, c and d are the lengths of sides opposite to vertices A, B, C and D respectively. Since, opposite angles of a cyclic quadrilateral are supplementary.**

** .........(1)**

**& **

**Now, in ΔABC and ΔACD we have **

** .........(2)**

**and {Cosine Rule}**

** {using (1)}**

** ........(3)**

**Now, using (2) and (3), we have **

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**B. Ptolemy's Theorem **

**In a cyclic quadrilateral, ABCD, we have {Cosine Rule}**

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**and **

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**Similarly, **

**Hence, **

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**C. Area of a Cyclic Quadrilateral**

**Area of Quadrilateral **

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**Since, opposite angles of cyclic quadrilateral are supplementary**

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**Also, **

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**Sequaring both sides, we have **

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**Substituting respective value in (1), we have**

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**Formula for the Circum-Radius of a Cyclic Quadrilateral**

**From the figure above, it is obvious that the circle circumscribing the quadrilateral ABCD is also the circum-circle of Let R be the radius of this circle and S be the area of quadrilateral ABCD. **

** ......(1)**

**Also, Area of Quadrilateral **

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**Since, opposite angles of cyclic quadrilateral are supplementary **

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** .......(2)**

**Using (1) and (2), we have**

** {using Ptolemy's Theorem}**

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