Circle Theorems

- proving chord of a circle

angles in a semicircle

The angle in a semicircle is a right angle.

proving angles in a semicircle

angle at the centre theorem

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

proving angle at the centre theorem

alternate segment theorem

the angle that lies between a tangent an a chord is equal to the angle subtended by the same chord in the alternate segment.

proving alternate segment theorem

angles in the same segment theorem

angles in the same segment are equal

chord of a circle

the perpendicular from the centre of a circle to a chord bisects the chord (splits the chord into two equal parts)

cyclic quadrilateral theorem

the opposite angles in a cyclic quadrilateral total 180 degrees.

tangent of a circle

(A) The angle between a tangent and radius is 90 degrees.

(B) Tangents which meet at the same point are equal in length.

tangents from a point