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Cyclic Quadrilateral: Properties, Theorems, and Examples

Definition of Cyclic Quadrilateral

A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle. In other words, its four vertices lie on a common circle, and the circle passes through all the vertices. The term “cyclic” comes from the Greek word “kyklos,” which means circle. Therefore, a cyclic quadrilateral is also known as a circumscriptible quadrilateral or an inscribed quadrilateral.

Cyclic quadrilaterals have some unique properties that distinguish them from other quadrilaterals. For example, the opposite angles of a cyclic quadrilateral are supplementary, which means that they add up to 180 degrees. Additionally, the sum of any two adjacent angles in a cyclic quadrilateral is also 180 degrees. These properties are based on the fact that the opposite sides of a cyclic quadrilateral are chords of the same circle, and any two adjacent chords in a circle intersect at a point called the midpoint of the arc that they define.

Properties of Cyclic Quadrilateral

Besides the properties mentioned above, cyclic quadrilaterals have some other interesting properties, which we will explore below.

1. The perpendicular bisectors of the sides of a cyclic quadrilateral intersect at a common point called the circumcenter. This point is equidistant from the four vertices of the quadrilateral and lies on the circumcircle.
2. The line joining the midpoints of the diagonals of a cyclic quadrilateral is perpendicular to the line joining the circumcenter and the centroid of the quadrilateral. The centroid is the point of intersection of the diagonals.
3. The product of the lengths of the diagonals of a cyclic quadrilateral is equal to the sum of the products of the opposite sides. In other words, if a cyclic quadrilateral has sides a, b, c, and d, and diagonals p and q, then p*q = ac + bd.
4. The area of a cyclic quadrilateral can be calculated using Brahmagupta’s formula, which states that the area is equal to the square root of (s-a)(s-b)(s-c)(s-d), where s is the semiperimeter of the quadrilateral, which is half the sum of its sides.
5. The circumradius of a cyclic quadrilateral can be calculated using the formula R = (abcd)/(4K), where R is the circumradius, a, b, c, and d are the sides of the quadrilateral, and K is its area.

Theorems of Cyclic Quadrilateral

Cyclic quadrilaterals also have some important theorems that can be used to solve problems related to them. Some of these theorems are listed below.

• Ptolemy’s theorem: This theorem states that in a cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of the opposite sides. In other words, if a cyclic quadrilateral has sides a, b, c, and d, and diagonals p and q, then p*q = ac + bd.
• Brahmagupta’s theorem:

  Brahmagupta’s theorem, also known as Brahmagupta’s formula, is a geometric theorem named after the ancient Indian mathematician Brahmagupta. The theorem relates to the area of a cyclic quadrilateral, which is a quadrilateral whose vertices all lie on a circle.

  According to Brahmagupta’s theorem, the area of a cyclic quadrilateral can be calculated using the following formula:

  Area = ?(s – a)(s – b)(s – c)(s – d)

  where s is the semiperimeter of the quadrilateral (i.e., half the sum of its four sides), and a, b, c, and d are the lengths of the four sides.

  This theorem is useful in solving problems involving cyclic quadrilaterals, which often appear in geometry and trigonometry. By using the theorem, it is possible to calculate the area of a cyclic quadrilateral based solely on the lengths of its sides, without needing to know the angles or any other information about the quadrilateral.