Circle Theorems (Higher Tier)

Key Definitions

A circle is a two-dimensional shape formed by all points equidistant from a single central point.
The radius of a circle is any line segment that connects the centre of the circle to any point on it.
A diameter is a line segment connecting two points on the circle which passes through the centre; hence, it is twice the length of the radius.
A chord is a line segment joining any two points on the circle.
A tangent is a straight line that touches the circle at a single point, called the point of contact.

Basic Circle Theorems

At the point where a tangent meets a circle, the angle it makes with the radius is always 90 degrees.
The angle in a semi-circle is a right angle. This means an angle drawn from each end of the diameter to the same point on the perimeter, forming a triangle, will always be a right angle.
The radii of a circle are always equal to each other, which is especially useful when working with polygons inside circles.
Chord properties: All angles inscribed in the segment opposite a particular chord are equal to half the angle subtended by the chord at the centre of the circle.

Advanced Circle Theorems

Angles at the centre and circumference: The angle subtended at the centre of a circle by two points on the circumference is twice the size of the angle subtended at the circumference by the same two points.
Angles in the same segment: Angles subtended by an arc (part of the circumference) or a chord at the circumference in the same segment are equal.
Opposite angles of a cyclic quadrilateral: Opposite angles in a quadrilateral inscribed in a circle add up to 180 degrees.
Alternate segment theorem: The angle between a tangent and a chord of a circle is equal to the angle subtended by the chord in the alternate segment.

Practicing these theorems through plenty of problems will not only help memorize the rules, but also understand their applications and their interactions with other geometrical properties.