Notes On Cyclic Quadrilaterals - CBSE Class 9 Maths
A circle can be drawn passing through three non-collinear distinct points. The points that lie on a circle are called concyclic points. So, three non-collinear points are always concyclic. 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 180°. If the sum of the opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic. The exterior angle formed by producing a side of a cyclic quadrilateral is equal to the interior opposite angle. A cyclic parallelogram is a rectangle.

#### Summary

A circle can be drawn passing through three non-collinear distinct points. The points that lie on a circle are called concyclic points. So, three non-collinear points are always concyclic. 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 180°. If the sum of the opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic. The exterior angle formed by producing a side of a cyclic quadrilateral is equal to the interior opposite angle. A cyclic parallelogram is a rectangle.

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