BodhiAI
Nov. 12, 2019

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

Since, opposite angles of cyclic quadrilateral are supplementary

Also,

Sequaring both sides, we have

Substituting respective value in (1), we have

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)

Since, opposite angles of cyclic quadrilateral are supplementary

.......(2)

Using (1) and (2), we have

{using Ptolemy's Theorem}