Cyclic Quadrilaterals

A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. This means that all four vertices of the quadrilateral touch the circumference of the circle. A key property of cyclic quadrilaterals relates to the angles formed by the vertices.

Formulae

The primary formula associated with cyclic quadrilaterals is the relationship between opposite angles:

**Sum of Opposite Angles:** In a cyclic quadrilateral, the sum of each pair of opposite angles is 180°.

Mathematically, for a cyclic quadrilateral ABCD:

$\angle A + \angle C = 180^\circ$
$\angle B + \angle D = 180^\circ$

Examples

Here are a few examples to illustrate the concept:

Example-1: Consider a cyclic quadrilateral ABCD. If $\angle A = 70^\circ$ and $\angle B = 110^\circ$, find $\angle C$ and $\angle D$.

Solution:

Since ABCD is cyclic, we know that $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$.

Therefore:

$\angle C = 180^\circ – \angle A = 180^\circ – 70^\circ = 110^\circ$
$\angle D = 180^\circ – \angle B = 180^\circ – 110^\circ = 70^\circ$

Example-2: In a cyclic quadrilateral PQRS, $\angle P = x + 20^\circ$, $\angle R = 2x – 10^\circ$. Find the values of $\angle P$ and $\angle R$.

Solution:

Since PQRS is cyclic, $\angle P + \angle R = 180^\circ$

Substituting the given values:

$(x + 20^\circ) + (2x – 10^\circ) = 180^\circ$

$3x + 10^\circ = 180^\circ$

$3x = 170^\circ$

$x = \frac{170}{3}^\circ$

Now, $\angle P = x + 20^\circ = \frac{170}{3}^\circ + 20^\circ = \frac{230}{3}^\circ \approx 76.67^\circ$

and $\angle R = 2x – 10^\circ = 2(\frac{170}{3}^\circ) – 10^\circ = \frac{340}{3}^\circ – \frac{30}{3}^\circ = \frac{310}{3}^\circ \approx 103.33^\circ$

Theorem with Proof

Theorem: The sum of opposite angles in a cyclic quadrilateral is 180 degrees.

Proof:

Let ABCD be a cyclic quadrilateral, and let O be the center of the circle. Join OA, OB, OC, and OD.

Let $\angle AOB = 2\alpha$, $\angle BOC = 2\beta$, $\angle COD = 2\gamma$, and $\angle DOA = 2\delta$.

Since the sum of angles around a point is $360^\circ$, we have:

$2\alpha + 2\beta + 2\gamma + 2\delta = 360^\circ$

$\alpha + \beta + \gamma + \delta = 180^\circ$

Now, consider the angles at the center and the circumference. For example, $\angle BCD$ is an inscribed angle subtended by the arc BAD. The central angle subtended by the same arc is the reflex angle BOD (which equals $2\alpha + 2\beta$)

Thus, $\angle BCD = \frac{1}{2} * (2\alpha + 2\gamma) = \alpha + \gamma$

Similarly, $\angle BAD = \frac{1}{2} * (2\beta + 2\delta) = \beta + \delta$

Therefore, $\angle BCD + \angle BAD = (\alpha + \gamma) + (\beta + \delta) = \alpha + \beta + \gamma + \delta = 180^\circ$

Similarly, we can prove that $\angle ABC + \angle ADC = 180^\circ$.

Hence, the sum of opposite angles in a cyclic quadrilateral is $180^\circ$.

Common mistakes by students

Confusing properties of cyclic quadrilaterals with other quadrilaterals: Students may incorrectly apply properties specific to other quadrilaterals (like parallelograms or rectangles) to cyclic quadrilaterals.
Incorrectly applying the converse: Students might struggle to correctly apply the converse of the theorem (If the sum of opposite angles of a quadrilateral is $180^\circ$, then the quadrilateral is cyclic.)
Forgetting to check if the quadrilateral is truly cyclic: Always verify that all vertices of a quadrilateral lie on a single circle before applying the cyclic quadrilateral properties.

Real Life Application

Cyclic quadrilaterals are indirectly used in many engineering and architectural designs, especially when circular elements are incorporated. For example:

Bridge Design: Arches in bridges often relate to cyclic quadrilaterals formed with other structural components.
Building Architecture: Designs with circular or semi-circular windows, domes or other circular elements can involve cyclic quadrilateral relationships.

Fun Fact

The properties of cyclic quadrilaterals were explored by ancient Greek mathematicians, and they are essential in geometry, offering elegant solutions to various problems.