1. What is a Cyclic Quadrilateral?
A cyclic quadrilateral is a quadrilateral whose all four vertices lie on a circle. This means the quadrilateral can be perfectly drawn inside a circle.
The circle that passes through all four vertices is called the circumcircle of the quadrilateral.
If points \(A, B, C, D\) lie on a circle with centre \(O\), then quadrilateral \(ABCD\) is cyclic.
2. Condition for a Quadrilateral to be Cyclic
Not every quadrilateral is cyclic. A quadrilateral becomes cyclic only when the opposite angles add up to \(180^\circ\).
2.1. Opposite Angle Property
\(\angle A + \angle C = 180^\circ\)
\(\angle B + \angle D = 180^\circ\)
If this condition holds, the quadrilateral must be cyclic.
2.2. Converse
If a pair of opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.
3. Properties of a Cyclic Quadrilateral
Cyclic quadrilaterals have some special angle properties that make them very useful in problem-solving.
3.1. Exterior Angle Property
The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
\(\text{Exterior angle at } A = \angle C\)
3.2. Equal Angles Subtend Equal Chords
Angles subtending the same chord in a cyclic quadrilateral are equal.
This is the same segment theorem applied inside the quadrilateral.
4. Cyclic Quadrilateral and Arcs
Cyclic quadrilaterals have strong relationships with the arcs they subtend.
4.1. Opposite Angles Subtend Supplementary Arcs
An opposite angle subtends an arc that is the complement of the arc subtended by the other opposite angle.
4.2. Chord–Angle Relationship
If chords \(AC\) and \(BD\) intersect inside the quadrilateral, the angle formed relates to arcs.
5. Examples for Better Understanding
These examples help visualise and apply the properties.
5.1. Example 1: Opposite Angles
If in quadrilateral \(ABCD\):
\angle A = 75^\circ, \; \angle C = 105^\circ
Then \(\angle A + \angle C = 180^\circ\). So the quadrilateral is cyclic.
5.2. Example 2: Find an Angle
In a cyclic quadrilateral, if:
\angle B = 120^\circ
Then the opposite angle is:
\angle D = 180^\circ - 120^\circ = 60^\circ
6. Real-Life Occurrences
Cyclic quadrilaterals often appear in:
- Architectural arches
- Round window frames
- Geometric art and mandala designs
- Engineering structures using circular joints