Theorem:
If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic).
Cyclic Quadrilateral:
A four-sided figure inscribed in a circle such that all its vertices lie on the circumference of the circle is said to be a cyclic quadrilateral.
Important!
- A quadrilateral are also called a \(4\)-gon.
- The angle subtended by the major arc at the centre \(O\) is called the reflex angle.
Theorem on a cyclic quadrilateral:
The sum of two opposite angles of a cyclic quadrilateral is \(180^{\circ}\).
Converse of Theorem: If two opposite angles of a quadrilateral add up to \(180°\), then the vertices of the quadrilateral lie on a circle, i.e., they are concyclic.
Important!
The exterior angle of a cyclic quadrilateral at a vertex is equal to the interior opposite angle.