• Sum of Opposite Angles: In a cyclic quadrilateral, the sum of each pair of opposite angles is $180^\circ$. That is, if the quadrilateral is $ABCD$, then $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$.
• Exterior Angle: An exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. For example, the exterior angle at vertex $A$ is equal to $\angle C$.
• Equal Chords Subtend Equal Angles: Chords of equal length in the circumcircle subtend equal angles at the center of the circle.
• Perpendicular Bisectors: The perpendicular bisectors of the sides of a cyclic quadrilateral intersect at the center of the circumcircle.

Ptolemy’s Theorem:

In a cyclic quadrilateral $ABCD$, the product of the diagonals is equal to the sum of the products of opposite sides. Mathematically, this is expressed as: $$AC \cdot BD = AB \cdot CD + AD \cdot BC$$

Where:

• $AC$ and $BD$ are the diagonals of the quadrilateral.
• $AB$, $BC$, $CD$, and $DA$ are the sides of the quadrilateral.

• Determining Side Lengths: If three sides and one diagonal of a cyclic quadrilateral are known, the fourth side can be determined using the theorem.
• Testing for Cyclicity: If the sides and diagonals of a quadrilateral satisfy Ptolemy’s equation, the quadrilateral is cyclic.
• Relation to Trigonometry: The theorem can be used in conjunction with trigonometric identities to solve for unknown angles and sides within the quadrilateral.

• Radius Determination: The radius of the circumcircle can be calculated using the formula: $$R = \frac{\sqrt{(ab + cd)(ac + bd)(ad + bc)}}{4K}$$ Where $a$, $b$, $c$, and $d$ are the sides of the quadrilateral, and $K$ is its area.
• Center of Circumcircle: The point where the perpendicular bisectors of the sides intersect is the center of the circumcircle.
• Inscribed Angles: Angles subtended by the same chord at the circumference are equal.

• Engineering: Used in the design of bridge structures and mechanical linkages.
• Architecture: Assists in creating aesthetically pleasing and structurally sound designs.
• Computer Graphics: Essential for rendering shapes and understanding spatial relationships in 3D modeling.

Example: Given a cyclic quadrilateral $ABCD$ with sides $AB = 5$, $BC = 6$, $CD = 5$, and $DA = 6$, and one of the diagonals $AC = 7$, find the length of the other diagonal $BD$.

Solution:

Using Ptolemy’s Theorem: $$AC \cdot BD = AB \cdot CD + AD \cdot BC$$ Substituting the known values: $$7 \cdot BD = 5 \cdot 5 + 6 \cdot 6$$ $$7 \cdot BD = 25 + 36$$ $$7 \cdot BD = 61$$ $$BD = \frac{61}{7} \approx 8.71$$

Proof:

Consider a cyclic quadrilateral $ABCD$ with vertices lying on a circle. Extend sides $AB$ and $CD$ to meet at point $E$. Since $ABCD$ is cyclic, angles $\angle A$ and $\angle C$ subtend the same arc. Therefore, $\angle A + \angle C = 180^\circ$ because they are supplementary angles.

Problem: In cyclic quadrilateral $ABCD$, $AB = 8$, $BC = 15$, $CD = 7$, and $DA = 10$. Find the lengths of the diagonals $AC$ and $BD$.

Solution:

Applying Ptolemy’s Theorem: $$AC \cdot BD = AB \cdot CD + AD \cdot BC$$ $$AC \cdot BD = 8 \cdot 7 + 10 \cdot 15$$ $$AC \cdot BD = 56 + 150$$ $$AC \cdot BD = 206$$

Additionally, using Brahmagupta’s Formula to find the area $K$:

Using the area formula related to diagonals: $$K = \frac{AC \cdot BD \cdot \sin \theta}{2}$$ Assuming $\theta$ is the angle between the diagonals, and since $ABCD$ is cyclic, $\theta = 90^\circ$, so $\sin \theta = 1$: $$88.36 = \frac{AC \cdot BD}{2}$$ $$AC \cdot BD = 176.72$$

However, from Ptolemy’s Theorem, $AC \cdot BD = 206$, which suggests a discrepancy. This indicates that the assumption $\theta = 90^\circ$ may not hold, and further trigonometric analysis is required to find the exact lengths of $AC$ and $BD$. This problem illustrates the need for integrating multiple concepts to solve advanced problems.

• Physics: Understanding the properties of cyclic quadrilaterals aids in analyzing forces and moments in mechanical systems.
• Engineering: Essential for designing components that require precise geometric arrangements.
• Computer Science: Utilized in algorithms for computer graphics and computational geometry.

Derivation of Ptolemy’s Theorem:

Consider a cyclic quadrilateral $ABCD$ inscribed in a circle. By extending sides $AD$ and $BC$, intersecting at point $E$, we can apply similar triangles and proportions to derive: $$AC \cdot BD = AB \cdot CD + AD \cdot BC$$ This derivation relies on properties of similar triangles and the power of a point theorem.

Problem: In cyclic quadrilateral $ABCD$, $AB = 10$, $BC = 6$, $CD = 8$, and $DA = x$. If one of the diagonals $AC = 12$, find the value of $x$.

Solution:

Applying Ptolemy’s Theorem: $$AC \cdot BD = AB \cdot CD + AD \cdot BC$$ $$12 \cdot BD = 10 \cdot 8 + x \cdot 6$$ $$12 \cdot BD = 80 + 6x$$

To find another equation, use Brahmagupta’s Formula to relate the sides and the radius. However, additional information is required to solve for $x$. This problem illustrates the complexity of advanced cyclic quadrilateral problems, often requiring multiple theorems and formulas for a solution.

• Cyclic quadrilaterals have all vertices lying on a single circle.
• Opposite angles sum to $180^\circ$, a key identifying property.
• Ptolemy’s Theorem and Brahmagupta’s Formula are essential for solving related problems.
• Advanced problem-solving often requires integrating multiple geometric concepts.
• Interdisciplinary applications highlight the importance of cyclic quadrilaterals beyond pure mathematics.