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BodhiAI
Nov. 12, 2019

Cyclic Quadrilateral:

 

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 

B. Ptolemy's Theorem 

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

and

Similarly,

Hence,

C. Area of a Cyclic Quadrilateral

Area of Quadrilateral

Since, opposite angles of cyclic quadrilateral are supplementary

Also,

Sequaring both sides, we have 

Substituting respective value in (1), we have

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

Since, opposite angles of cyclic quadrilateral are supplementary 

 

 

 

  .......(2)

Using (1) and (2), we have

  {using Ptolemy's Theorem}