You can draw a circle passing through three non-collinear distinct points. The points that lie on a circle are called concyclic points. So we can say three non-collinear points are always concyclic.
Theorem: If a line segment joining two points subtends equal angles at two other points on the same side of the line segment then all the four points are concyclic.
Given: Line segment AB.
Mark two points C and D such that ACB = ∠ADB.
To prove: A, B, C and D are concyclic points.
Draw a circle through points A, B and C.
Assume that the circle drawn through points A, B and C does not pass through D, and intersects AD at D’.
Proof: If A, B, C and D’ are concyclic:
∠ACB = ∠AD’B (Angles subtended by a chord in the same segment of a circle)
∠ACB = ∠ADB (Given)
∴ ∠AD’B = ∠ADB
Or D’ coincides with D.
Thus, A, B, C and D are concyclic points.
Hence, the theorem is proved.
A quadrilateral whose vertices lie on a circle is called a cyclic quadrilateral.
In a cyclic quadrilateral, the sum of the opposite angles is always equal to 180o.
If the sum of the opposite angles of a quadrilateral is 180o, then the quadrilateral is cyclic.