Circle Theorems and Proofs

Chord

Centre

Radius

Circumference

Arc

Diameter

Tangent

Sector

Segment

Angle in a semicircle is 90 degrees

Angle in a semicircle is 90 degrees PROOF

OA = OB = OC (all radii)

Angle OAC =x because base angles in an isosceles triangle are equal

Angle COA = 180-2x because angles in triangle add up to 180^o.

Opposite angles of a cyclic quadrilateral add to 180 degrees

Opposite angles of a cyclic quadrilateral add to 180 degrees PROOF

Angles in the same segment are equal

Angles in the same segment are equal PROOF

Angle at the centre is twice angle at the circumference

Angle at the centre is twice angle at the circumference PROOF

Angle between a radius & tangent is 90 degrees

Alternate segment theorem

Alternate segment theorem PROOF

Tangents from the same point are the same length

The perpendicular from the centre of the circle bisects the chord