A quadrilateral in a circle is something interesting in geometry. It plays an important role in large areas of mathematics, especially in geometry.

The quadrilateral is a geometrical shape that has four sides, four vertices, and four angles. They can be classified into various types based on their properties. But in the cyclic quadrilateral all the four vertices lie on the circumference of the circle.

In this, a detailed view of what a cyclic quadrilateral is, stating its properties and its applications.

A cyclic quadrilateral is a four-sided polygon whose vertices lie on one single circle. The circle in which the vertices lie is usually referred as the circumcircle of the quadrilateral. Its distinguishing property leads to some important and useful formulae by which the cyclic quadrilaterals are analyzed with ease in geometry.

Before understanding the formulas, let's first revise the key properties of cyclic quadrilateral:

Opposite Angles Supplementary: The opposite angles of a cyclic quadrilateral are supplementary. More precisely, any pair of opposite angles in a cyclic quadrilateral is always additive to 180 degrees.

For example:

Let ∠P, ∠Q, ∠R, and ∠S be the four angles of an inscribed quadrilateral. Then,

∠P + ∠R = 180°

∠Q + ∠S = 180°

Hence, an inscribed quadrilateral meets angle sum property of a quadrilateral, i.e the sum of all angles equals 360 degrees. Thus,

∠P + ∠Q+ ∠R+ ∠S = 360°

Angles Subtended By The Same Arc: Angles subtended by the same arc are equal. This means that if two angles in a circle intercept the same arc, they will have the same measures.

For example: If two angles ∠A and ∠B are subtended by the same arc in a circle, then:

∠A=∠B

The formula for the radius of a cyclic quadrilateral is:

where,

R is the radius of the circle circumscribing the quadrilateral

a, b, c, and d are the lengths of the sides of the quadrilateral

s is the semi-perimeter of the quadrilateral.

Diagonals Of Cyclic Quadrilateral

Suppose p,q,r and s are the sides of a cyclic quadrilateral and a and b are the diagonals, then we can find the diagonals of it using the below-given formulas:

Formula: The area K of a cyclic quadrilateral can be calculated using Brahmagupta’s formula. If a,b,c, and d are the lengths of the sides of the cyclic quadrilateral, and sss is the semiperimeter given by:

then the area K is given by:

ACBD=ABCD+BCAD

Theorem Application: Ptolemy's Theorem is useful for solving problems dealing with the diagonals of a cyclic quadrilateral when the lengths of the sides are known. It can find the length of a diagonal or can verify whether the given quadrilateral is cyclic or not.

Cyclic quadrilaterals possess a vast field of research in geometry, where several key formulas exist that make it much easier to conduct analyses and find solutions to related problems. From the area formula to Ptolemy's theorem for diagonals, each formula is a different tool to help students in their work with cyclic quadrilaterals. Their mastery will bring about not only a significant enhancement in your geometric problem-solving skills but also a solid foundation for understanding more complex geometric conceptions.

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Formula: Ptolemy’s Theorem relates the sides and diagonals of a cyclic quadrilateral. For a cyclic quadrilateral ABCD with diagonals AC and BD, the theorem states: