# cyclic quadrilateral

## Derivation / Proof of Ptolemy's Theorem for Cyclic Quadrilateral

Ptolemy's theorem for cyclic quadrilateral states that the product of the diagonals is equal to the sum of the products of opposite sides. From the figure below, Ptolemy's theorem can be written as

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## Derivation of Formula for Area of Cyclic Quadrilateral

For a cyclic quadrilateral with given sides a, b, c, and d, the formula for the area is given by

Where s = (a + b + c + d)/2 known as the semi-perimeter.

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## Quadrilateral with one side as diameter of circumscribing circle

**Problem PG-010**

The quadrilateral ABCD shown in Fig. PG-010 is inscribed in a circle with side AD coinciding with the diameter of the circle. if sides AB, BC, and CD are 8 cm, 10 cm, and 12 cm long, respectively, find the radius of the circumscribing circle.

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## The Cyclic Quadrilateral

A quadrilateral is said to be cyclic if its vertices all lie on a circle. In cyclic quadrilateral, the sum of two opposite angles is 180° (or π radian); in other words, the two opposite angles are supplementary.

$B + D = 180^\circ$

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## The Quadrilateral

Quadrilateral is a polygon of four sides and four vertices. It is also called tetragon and quadrangle. In the triangle, the sum of the interior angles is 180°; for quadrilaterals the sum of the interior angles is always equal to 360°

**Classifications of Quadrilaterals**

There are two broad classifications of quadrilaterals; *simple* and *complex*. The sides of simple quadrilaterals do not cross each other while two sides of complex quadrilaterals cross each other.

Simple quadrilaterals are further classified into two: *convex* and *concave*. Convex if none of the sides pass through the quadrilateral when prolonged while concave if the prolongation of any one side will pass inside the quadrilateral.

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