circle theorem

In geometry, the relationship between lines and angles within a circle. These can be used in further geometrical proofs.

1. If AB is the diameter of a circle and C is a point on the circumference of the circle, then angle ACB = 90°. The diameter AB subtends (forms) the angle ACB. Similarly, angle ABC stands on the arc AC, or AC subtends an angle ABC at B and angle BAC stands on the arc BC, or BC subtends an angle BAC at A.

2. A quadrilateral inscribed within a circle is called a cyclic quadrilateral and its opposite angles add up to 180°.

The exterior angle of a cyclic quadrilateral is equal to the opposite angle.

3. If AB is a chord of a circle with centre O and C is a point on the circumference of the circle in the same segment as O, then angle AOB = 2 × angle ACB. (The angle at the centre is twice the angle at the circumference.)

4. If AB is a chord of a circle and C and D are points in the same segment, then angle ACB = angle ADB. (Angles in the same segment are equal.)

5. A line meeting a circle at one point only is a tangent to the circle and is perpendicular to the radius at that point.

6. If point P lies outside a circle, then the two tangents from P to the circle are equal in length.

7. If AB and CD are chords of a circle meeting at X either inside or outside the circle, then XA × XB = XC × XD.