1. Circle Theorems Revision

2. 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

3. 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

4. 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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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