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**Circle Theorems Revision**

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Circle Theorem 1 The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc. AOB = 2x ACB A B O C

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Circle Theorem 2 Every angle at the circumference of a semicircle that is subtended by the diameter of the semicircle is a right angle. A B C D

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Circle Theorem 3 Angles at the circumference in the same segment of a circle are equal. Points on the circumference are subtended by the same arc AB. AC1B = AC2B A B C2 C1 C3

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Circle Theorem 4 The sum of the opposite angles of a cyclic quadrilateral is 180o. A B D C

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Circle Theorem 5 A tangent to a circle is perpendicular to the radius drawn to the point of contact. OX is perpendicular to AB X B O A

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Circle Theorem 6 Tangents to a circle from an external point to the point of contact are equal in length. AX = AY A Y X

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Circle Theorem 7 The line joining an external point to the centre of the circle bisects the angle between the tangents. OAX = OAY A Y X O

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**Circle Theorem 8 A radius bisects a chord at 90o.**

If O is the centre of the circle BMO = 90o, BM = CM B M O C

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Circle Theorem 9 The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. PTA = TBA A T B Q P

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A B O C A B C D A B C2 C1 C3 A B D C 1 A Y X 2 3 A Y X O 4 X B O A 6 7 5 A T B Q P B M O C 8 9

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