MATH circle theorems

Chord theorem 1

chords of equal length subtend equal angles at the centre of the circle, converse- if equal angles at centre subtend chords, they are equal

Chord theorem 2

chords of equal length are equidistant from the centre of the circle, converse- if chords are equidistant from centre, they are equal

Chord theorem 3

The perpendicular from the centre to the chord bisects the chord and the angle at the centre , converse- if a radius bisects the chord (or angle at the centre), then it is perpendicular to the chord.

Chord theorem 4

perpendicular bisectors of every chord in a circle intersect at the centre

Circle theorem 1: Angle at the centre and circumference

the angle at the centre of a circle is twice the angle at a point of the circumference

Circle theorem 2: Angle in a semicircle

The angle in a semicircle is 90∘

Circle theorem 3: Angles at the circumference

Angles at the circumference of a circle subtended by the same arc are equal.

Circle theorem 4: Opposite angles in cyclic quadrilaterals

Opposite angles in a cyclic quadrilateral are supplementary (sum to 180∘).

Tangents

tangent is perpendicular to the radius at the point of contact, two different tangents drawn from an external point to the circle creates line segments of equal length

alternate segment theorem

The angle between a tangent and a chord is equal to the angle in the alternate segment.