circle theorem

• circle theorems

  
• Rule 1

  Angles in the same segment are equal.

  x =x 

  Triangles drawn from the same chord will have the same angle when touching the circumference.

  
• Rule 2

  Opposite angles in a cyclic quadrilateral add up to  180°180°.

  w + x = 180°

  y + z = 180°

  This is a 4 sided shape with every corner touching the circumference of the circle.

  
• Rule 3

  The angle at the centre is twice the angle at the circumference. 

  The angle formed at the centre is exactly twice the angle at the circumference of a circle.

  
• Rule 4

  The perpendicular bisector of a chord passes through the centre of the circle. 

  A line perpendicular and in the centre of a chord (a line drawn across the circle) will always pass through the centre of the circle.

  
• Rule 5

  • The radius will always meet a tangent to the circle at 90°
  • A tangent (a line touching a single point on the circumference) will always make an angle of exactly 90° with the radius.
  • You can say that a tangent and radius that meet are perpendicular to each other.

  
• Rule 6

  The tangents from the same point to a circle are equal in length.

  AB = BC

  Two tangents (a line touching a single point on the circumference) drawn from the same outside point are always equal in length.

  
• Rule 7

  The angle inscribed in a semicircle is always a right angle. 

  A triangle drawn with the diameter will always make a 90°90° angle where it hits the circumference.

  Another way of saying this is that a diameter ‘subtends’ a right-angle at the circumference.

  
• Rule 8

  Alternate Segment Theorem: The angle between the tangentand the side of the triangle is equal to the opposite interior angle.

  x = x

  y = y

  The angle between the tangent and the triangle will be equal to the angle in the alternate segment.

  (This is the hardest rule and can be tricky to spot).