Circle definitions, properties and core theorems
Geometry and measures • Properties and constructions
Flashcards
Test your knowledge with interactive flashcards
Key concepts
What you'll likely be quizzed about
Basic definitions
Centre: a point equidistant from all points on the circle. Radius: a line segment from the centre to any point on the circle. Diameter: a chord passing through the centre; length equals twice the radius. Chord: a line segment with endpoints on the circle. Circumference: the distance around the circle; proportional to the radius (C = 2πr). Arc: a continuous part of the circumference between two points. Sector: the region bounded by two radii and the connecting arc. Segment: the region bounded by a chord and the corresponding arc. Each definition limits which constructions and theorems apply; for example, diameters always pass through the centre while chords may not.
Radius and tangent relationship
A radius drawn to the point of contact with a tangent is perpendicular to the tangent. Cause: the tangent touches the circle at exactly one point. Effect: the radius at that point meets the tangent at a right angle. Consequence: a line claimed to be tangent can be tested by checking perpendicularity with the radius to the contact point; conversely, perpendicularity implies tangency at that point.
Perpendicular from centre to chord
If a radius or diameter is perpendicular to a chord, then it bisects the chord and the subtended arcs. Cause: two radii to the chord's endpoints form an isosceles triangle. Effect: perpendicular from the vertex (centre) splits the base (chord) into equal parts. Consequence: perpendicular bisector of a chord passes through the centre; this gives a method to locate the centre from a chord.
Equal chords and distances from centre
Equal chords subtend equal arcs and lie at equal distances from the centre. Cause: equal chords create equal isosceles triangles with radii. Effect: identical vertex angles and identical perpendicular distances from centre to each chord. Consequence: chord length comparisons lead to centre-to-chord distance comparisons and vice versa.
Angle in a semicircle (Thales' theorem)
An angle subtended by a diameter at any point on the circle is a right angle. Cause: endpoints of the angle join to form a triangle whose base is a diameter. Effect: two radii to the base endpoints produce an isosceles decomposition that forces the remaining angle to be 90°. Consequence: any triangle drawn with one side as the diameter is a right triangle.
Angle at the centre and at the circumference
An angle at the centre is twice any angle subtended by the same arc at the circumference. Cause: the central angle spans the whole arc while the inscribed angle spans the same arc from the circumference. Effect: central angle = 2 × inscribed angle subtending the same arc. Consequence: knowing a central angle gives an inscribed angle and vice versa; this supports solving angle-chasing problems.
Angles in the same segment
Angles in the same segment, subtended by the same chord, are equal. Cause: each inscribed angle intercepts the same arc and therefore uses the same fraction of the arc's measure. Effect: equal inscribed angles at different points on the same arc. Consequence: angle equality can identify parallel lines or deduce other angles inside the circle.
Tangent-chord theorem (angle between tangent and chord)
The angle between a tangent and a chord through the point of contact equals the angle in the opposite arc subtended by the chord. Cause: draw the radius to the point of contact and use the central–inscribed angle relationship. Effect: tangent-chord angle equals the inscribed angle on the far side of the chord. Consequence: tangents provide external angle information that links to interior inscribed angles.
Cyclic quadrilaterals
Opposite angles of a quadrilateral inscribed in a circle sum to 180°. Cause: each opposite angle subtends supplementary arcs that together cover the full circumference. Effect: angleA + angleC = 180° and angleB + angleD = 180°. Consequence: angle sums give a route for proofs involving four points on a circle and for solving unknown angles.
Proof strategies using circle theorems
Common proof steps: identify radii to construct isosceles triangles, use perpendiculars to bisect chords, apply central–inscribed angle relation, employ tangent perpendicularity. Cause: constructing radii and drawing auxiliary lines reveals known triangle types. Effect: angle chasing and congruence arguments produce short rigorous proofs. Consequence: complex results reduce to combinations of the basic theorems above.
Key notes
Important points to keep in mind