Circle proofs draw on a fixed toolbox of theorems. Each row below gives the statement together with a standard reason phrasing; equivalent mathematically precise wording is also valid. In every diagram, \(O\) is the centre of the circle. Symmetry and tangent properties PropertyStatement and reasonDiagram Perpendicular bisector of a chordThe perpendicular from the centre bisects the chord, and the perpendicular bisector of a chord passes through the centre. If \(OM\perp AB\), then \(AM=BM\). (\(\perp\) bisector of chord) Equal chordsEqual chords are equidistant from the centre. If \(AB=CD\), then \(OM=ON\). (equal chords, equidistant from centre) Tangent and radiusA tangent is perpendicular to the radius drawn to the point of contact. \(OA\perp PQ\) (tangent \(\perp\) radius) Tangents from an external pointTangents from an external point are equal in length, and the line to the centre bisects the angle between them. \(PA=PB\) (tangents from an external point) Angle properties PropertyStatement and reasonDiagram Angle at the centreThe angle at the centre is twice any angle at the circumference standing on the same arc. \(\angle AOB=2\angle ACB\) (\(\angle\) at centre \(=2\angle\) at circumference) Angle in a semicircleIf \(AB\) is a diameter, the angle at the circumference is a right angle. \(\angle ACB=90^\circ\) (\(\angle\) in semicircle) Angles in the same segmentAngles standing on the same chord, on the same side of it, are equal. \(\angle ACB=\angle ADB\) (\(\angle\)s in the same segment) Angles in opposite segmentsOpposite angles of a cyclic quadrilateral are supplementary. \(a+b=180^\circ\) (\(\angle\)s in opposite segments) Alternate segment theoremThe angle between a tangent and a chord equals the angle in the alternate segment. \(\angle PAB=\angle ACB\) (Alternate Segment Theorem) Exam check A right angle at the circumference needs the chord to be a diameter — check before quoting (\(\angle\) in semicircle). For the alternate segment theorem, name the tangent and the chord, then take the equal angle on the other side of the chord.

Circle Theorems

Secondary 4

TGM Original Questions

Circle proofs draw on a fixed toolbox of theorems. Each row below gives the statement together with a standard reason phrasing; equivalent mathematically precise wording is also valid. In every diagram, \(O\) is the centre of the circle.

Symmetry and tangent properties

Property	Statement and reason	Diagram
Perpendicular bisector of a chord	The perpendicular from the centre bisects the chord, and the perpendicular bisector of a chord passes through the centre. If \(OM\perp AB\), then \(AM=BM\). (\(\perp\) bisector of chord)
Equal chords	Equal chords are equidistant from the centre. If \(AB=CD\), then \(OM=ON\). (equal chords, equidistant from centre)
Tangent and radius	A tangent is perpendicular to the radius drawn to the point of contact. \(OA\perp PQ\) (tangent \(\perp\) radius)
Tangents from an external point	Tangents from an external point are equal in length, and the line to the centre bisects the angle between them. \(PA=PB\) (tangents from an external point)

Angle properties

Property	Statement and reason	Diagram
Angle at the centre	The angle at the centre is twice any angle at the circumference standing on the same arc. \(\angle AOB=2\angle ACB\) (\(\angle\) at centre \(=2\angle\) at circumference)
Angle in a semicircle	If \(AB\) is a diameter, the angle at the circumference is a right angle. \(\angle ACB=90^\circ\) (\(\angle\) in semicircle)
Angles in the same segment	Angles standing on the same chord, on the same side of it, are equal. \(\angle ACB=\angle ADB\) (\(\angle\)s in the same segment)
Angles in opposite segments	Opposite angles of a cyclic quadrilateral are supplementary. \(a+b=180^\circ\) (\(\angle\)s in opposite segments)
Alternate segment theorem	The angle between a tangent and a chord equals the angle in the alternate segment. \(\angle PAB=\angle ACB\) (Alternate Segment Theorem)

Exam check

A right angle at the circumference needs the chord to be a diameter — check before quoting (\(\angle\) in semicircle). For the alternate segment theorem, name the tangent and the chord, then take the equal angle on the other side of the chord.

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