Cyclic Quadrilateral Formula

Cyclic Quadrilateral Formula

The Cyclic Quadrilateral Formula is a four-sided polygon encircled by a circle. With the given side lengths, it has the maximum area possible. With those side lengths, a quadrilateral inscribed in a circle illustrates the maximum area possible. In this article, one can explore the properties of a Cyclic Quadrilateral Formula.

Cyclic Quadrilateral Definition

The Cyclic Quadrilateral Formula is a quadrilateral encircled by a circle. All four vertices of the quadrilateral are connected by a circle. Concyclic vertices are said to be concentric. Circumcenter and circumradius refer to the centre and radius of a circle, respectively.

‘Cyclic’ comes from the Greek word ‘kuklos’, which means ‘circle’ or ‘wheel’. Quadrilaterals are derived from the Latin word quadri, which means “four-sided” or “latus”.

ABCD is a cyclic quadrilateral with side lengths a, b, c, and d, and diagonals p and q.

Properties of Cyclic Quadrilateral

We can identify cyclic quadrilaterals easily and solve questions based on them by knowing their properties. The following are some properties of a Cyclic Quadrilateral Formula:

• The four vertices of a cyclic quadrilateral lie on the circle’s circumference.
• Circles have four chords, which are the four sides of the inscribed quadrilateral.
• An exterior angle at a vertex equals its opposite interior angle.
• In a cyclic quadrilateral, p × q is the sum of the products of opposite sides, where p and q represent the diagonals.
• There is always a concurrent bisector between the perpendicular lines.
• At the centre O, the perpendicular bisectors of the four sides of the Cyclic Quadrilateral Formula meet.
• A pair of opposite angles add up to 180° (supplementary). In other words, A+C and B+D equal 180°.

Area of a Cyclic Quadrilateral

A cyclic quadrilateral’s area can be calculated as follows:

√(s−a) (s−b) (s−c) (s−d)

Assuming that “s” is the semi-perimeter,

s = a + b +c + d / 2

Theorems on Cyclic Quadrilateral

In addition to the property of Cyclic Quadrilateral Formula that the sum of opposite angles is always 180°, there are two other theorems on cyclic quadrilaterals. The following are explained:

Ptolemy Theorem of Cyclic Quadrilateral

In a Cyclic Quadrilateral Formula with successive vertices A, B, C, D, sides a = AB, b = BC, c = CD, d = DA, and diagonals p = AC, q = BD, according to the Ptolemy theorem. Diagonals can be expressed as p × q = (a × c) + (b × d).

Brahmagupta Theorem of Cyclic Quadrilateral

Based on Brahmagupta’s theorem, a Cyclic Quadrilateral Formula with perpendicular diagonals always has a perpendicular to a side that bisects its opposite side. It was named after the Indian mathematician Brahmagupta (598-668).

Let A, B, C and D be four points on a circle perpendicular to AC and BD. The intersection of AC and BD is indicated by M. The intersection of M and BC is called E. The line EM intersects the edge AD. F is then the midpoint of AD according to the theorem. It is necessary to prove that AF = FD. It will be shown that both AF and FD are equal to FM.

First, note that FAM and CBM are inscribed angles that intercept the same arc of the circle, so AF = FM. Additionally, CBM and CME are both complementary to BCM (i.e., they add up to 90°), and are therefore equal. Finally, CME and FMA have the same angle. Thus, AFM is an isosceles triangle with equal sides AF and FM. The proof that FD = FM goes similarly: FDM, BCM, BME, and DMF are all equal angles, so DFM is an isosceles triangle, so FD = FM. According to the theorem, AF = FD.

Related Articles on Cyclic Quadrilateral

• The indenters M 1, M 2, M 3, M 4 in triangles DAB, ABC, BCD, and CDA are the vertices of a rectangle in a cyclic quadrilateral ABCD. The Japanese theorem is one of those theorems. There are four triangles with orthocenters congruent to ABCD, and there are four triangles with centroids congruent to another Cyclic Quadrilateral Formula.
• Let P be the point where the diagonals AC and BD intersect in a Cyclic Quadrilateral Formula ABCD. In this case, the angle APB is the arithmetic mean of the angles AOB and COD. The inscribed angle theorem and the exterior angle theorem lead directly to this conclusion.
• A Cyclic Quadrilateral Formula with the rational area and unequal rational sides is not found in either arithmetic or geometric progression.

• It is also ex-bicentric if the side lengths of a Cyclic Quadrilateral Formula form an arithmetic progression.
• The internal angle bisectors of the angles at E and F are perpendicular if the opposite sides of a Cyclic Quadrilateral Formula meet at E and F.

Cyclic Quadrilateral Examples

In the summation of the angles of consecutive angles is equal if and only if the spherical quadrilateral formed by four intersecting greater circles is cyclic. Therefore, α + γ = β + δ for consecutive angles α, β, γ, δ of the quadrilateral. The sums of opposite angles in a spherical quadrilateral inscribed in a small circle of a sphere are equal, and the sums of opposite sides in a circumscribed quadrilateral are equal, according to Lexell. This theorem is the spherical equivalent of the plane theorem, and the second theorem is its dual, i.e., the result of switching great circles and their poles. In spherical quadrilaterals, Kiper et al. proved the converse of this theorem: If the sums of the opposite sides are equal, then the quadrilateral has an inscribing circle.

Practice Questions on Cyclic Quadrilateral

1. Example 1: In the given problem, PQSR is a Cyclic Quadrilateral Formula, and △ PQR is an equilateral triangle, then find the measure of ∠ QSR.
   Solution: Given, △ PQR is an equilateral triangle. ∴ ∠ QPR=60°. Angles opposite each other are supplementary.
   ∴ ∠ QSR + ∠ QPR = 180°
   From the above two equations we have,
   ∠ QSR + 60° = 180°
   ∠ QSR = 180° − 60°
   ∠ QSR = 120°
   ∴ The measure of ∠ QSR = 120°.