Circle theorems

1. Parts of a circle

Radius: centre to edge. Diameter ($2\times$ radius): edge to edge through the centre. Circumference: the perimeter. Chord: a line joining two points on the edge, not through the centre. Tangent: a line that touches the circle at exactly one point.

2. Tangent and radius

A tangent and the radius drawn to the point of contact always meet at $90°$.

3. Isosceles triangle

Any triangle formed by two radii is isosceles, because both radii have the same length.

4. Perpendicular bisector of a chord

A line from the centre that cuts a chord exactly in half always meets it at $90°$.

5. Angle at the centre

The angle at the centre is always twice the angle at the circumference, when both are drawn from the same arc.

$a = 2b$

6. Angle in a semicircle

A triangle drawn from both ends of a diameter to any point on the circumference always has a $90°$ angle at that point.

7. Angles in the same segment

Angles drawn from the same arc or chord, on the same side, are always equal. In a cyclic quadrilateral, opposite angles always add up to $180°$.

8. Tangents from a point

Two tangents drawn to a circle from the same external point are always equal in length.

$PA = PB$

9. Alternate segment theorem

The angle between a tangent and a chord always equals the angle in the alternate segment (the angle made by that chord on the other side of the circle).