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Cyclic Quadrilaterals

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Cyclic Quadrilaterals - Lesson 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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