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3 Unit Maths – Circle Geometry
Terms
Chord
A chord is a line segment joining any two points on the circle.
 Divides circle into two parts, the minor arc and major arc.
Secant
A secant is a line that goes through a circle.
Sector
A sector is the region between two radii and the arc cut off by these radii.
Segment
A segment is the region between a chord and the arc cut off by the chord.
Cyclic quadrilateral
A cyclic quadrilateral is a quadrilateral inscribed in a circle.
The circle is circumscribed about the quadrilateral.
Concyclic points
Concyclic points are points that belong to the same circle.
Concentric circles
Concentric circles are circles that have the same centre.
Theorems
Equal angles at the centre subtend equal arcs, and vice versa
𝑖𝑓 ∠𝐴𝑂𝐵 = ∠𝐶𝑂𝐷 𝑡ℎ𝑒𝑛 𝑎𝑟𝑐 𝐴𝐵 = 𝑎𝑟𝑐 𝐶𝐷
𝑖𝑓 𝑎𝑟𝑐 𝐴𝐵 = 𝑎𝑟𝑐 𝐶𝐷 𝑡ℎ𝑒𝑛 ∠𝐴𝑂𝐵 = ∠𝐶𝑂𝐷
Equal chords subtend equal arcs, and vice versa
𝑖𝑓 𝐴𝐵 = 𝐶𝐷 𝑡ℎ𝑒𝑛 𝑎𝑟𝑐 𝐴𝐵 = 𝑎𝑟𝑐 𝐶𝐷
𝑖𝑓 𝑎𝑟𝑐 𝐴𝐵 = 𝑎𝑟𝑐 𝐶𝐷 𝑡ℎ𝑒𝑛 𝐴𝐵 = 𝐶𝐷
Equal chords are equidistant from the centre, and vice versa
𝐸 𝑎𝑛𝑑 𝐹 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐ℎ𝑜𝑟𝑑𝑠 𝐴𝐵 𝑎𝑛𝑑 𝐶𝐷 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝑖𝑓 𝐴𝐵 = 𝐶𝐷 𝑡ℎ𝑒𝑛 𝑂𝐸 = 𝑂𝐹
𝑖𝑓 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑏𝑖𝑠𝑒𝑐𝑡𝑜𝑟𝑠 𝑂𝐸 = 𝑂𝐹 𝑡ℎ𝑒𝑛 𝐴𝐵 = 𝐶𝐷
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3 Unit Maths – Circle Geometry
The perpendicular bisector of a chord passes through the centre, and vice versa
𝑖𝑓 𝐸 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐵 𝑎𝑛𝑑 𝑂 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑒, 𝑡ℎ𝑒𝑛 𝑂𝐸 ⊥ 𝐴𝐵
𝑖𝑓 𝑂𝐸 ⊥ 𝐴𝐵, 𝑤ℎ𝑒𝑟𝑒 𝑂 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑒 𝑡ℎ𝑒𝑛 𝐸 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐵
𝑖𝑓 𝑂𝐸 ⊥ 𝐴𝐵, 𝑤ℎ𝑒𝑟𝑒 𝐸 𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐵 𝑡ℎ𝑒𝑛 𝑂 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑒
The angle at the centre is twice the angle at the circumference subtended by the same arc
∠𝐴𝑂𝐵 = 2∠𝐴𝑃𝐵
Angles subtending the same arc are equal
𝑖𝑓 𝑃, 𝑄, 𝐴, 𝐵 𝑎𝑟𝑒 𝑐𝑜𝑛𝑐𝑦𝑐𝑙𝑖𝑐 𝑡ℎ𝑒𝑛 ∠𝐴𝑃𝐵 = ∠𝐴𝑄𝐵
𝑖𝑓 ∠𝐴𝑃𝐵 = ∠𝐴𝑄𝐵 𝑡ℎ𝑒𝑛 𝑃, 𝑄, 𝐴, 𝐵 𝑎𝑟𝑒 𝑐𝑜𝑛𝑐𝑦𝑐𝑙𝑖𝑐
The angle on a semi-circle is a right angle, and vice versa
𝑖𝑓 𝐴𝐵 𝑖𝑠 𝑎 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑡ℎ𝑒𝑛 ∠𝐴𝑃𝐵 = 90°
𝑖𝑓 ∠𝐴𝑃𝐵 = 90° 𝑡ℎ𝑒𝑛 𝐴𝐵 𝑖𝑠 𝑎 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟
The opposite angles of a cyclic quadrilateral are supplementary, and vice versa
𝑖𝑓 𝐴, 𝐵, 𝐶, 𝐷 𝑎𝑟𝑒 𝑐𝑜𝑛𝑐𝑦𝑐𝑙𝑖𝑐 𝑡ℎ𝑒𝑛 ∠𝐴𝐵𝐶 + ∠𝐴𝐷𝐶 = 180°
𝑖𝑓 ∠𝐴𝐵𝐶 + ∠𝐴𝐷𝐶 = 180° 𝑡ℎ𝑒𝑛 𝐴, 𝐵, 𝐶, 𝐷 𝑎𝑟𝑒 𝑐𝑜𝑛𝑐𝑦𝑐𝑙𝑖𝑐
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3 Unit Maths – Circle Geometry
The exterior angle of a cyclic quadrilateral equals the opposite interior angle, and vice versa
𝑖𝑓 𝐴, 𝐵, 𝐶, 𝐷 𝑎𝑟𝑒 𝑐𝑜𝑛𝑐𝑦𝑐𝑙𝑖𝑐 𝑡ℎ𝑒𝑛 ∠𝐴𝐵𝐶 = ∠𝐶𝐷𝑋
𝑖𝑓 ∠𝐴𝐵𝐶 = ∠𝐶𝐷𝑋 𝑡ℎ𝑒𝑛 𝐴, 𝐵, 𝐶, 𝐷 𝑎𝑟𝑒 𝑐𝑜𝑛𝑐𝑦𝑐𝑙𝑖𝑐
The tangent to a circle is perpendicular to the radius drawn to the point of contact, and vice versa
𝑇 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑐𝑜𝑛𝑡𝑎𝑐𝑡 𝑜𝑓 𝑋𝑌 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒
𝑖𝑓 𝑂𝑇 ⊥ 𝑋𝑌 𝑡ℎ𝑒𝑛 𝑋𝑌 𝑖𝑠 𝑎 𝑡𝑎𝑛𝑔𝑒𝑛𝑡
𝑖𝑓 𝑋𝑌 𝑖𝑠 𝑎 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡ℎ𝑒𝑛 𝑂𝑇 ⊥ 𝑋𝑌
Tangents to a circle from an external point are equal
𝑖𝑓 𝑃𝐴 𝑎𝑛𝑑 𝑃𝐵 𝑎𝑟𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑠 𝑡ℎ𝑒𝑛 𝑃𝐴 = 𝑃𝐵
Alternate segment theorom: the angle between a tangent and a chord through the point of contact is
equal to the angle in the alternate segment
𝐼𝑓 𝑋𝑌 𝑖𝑠 𝑎 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡ℎ𝑎𝑡 𝑡𝑜𝑢𝑐ℎ𝑒𝑠 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝑖𝑛 𝐴 𝑡ℎ𝑒𝑛 ∠𝐴𝑃𝐵 = ∠𝑋𝐴𝐵
Intersecting chord theorem: if two chords intersect in a circle the product of the segments of one chord
equals the product of segments of the other
𝑖𝑓 𝑡𝑤𝑜 𝑐ℎ𝑜𝑟𝑑𝑠 𝐴𝐵 𝑎𝑛𝑑 𝐶𝐷 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡 𝑎𝑡 𝑋 𝑡ℎ𝑒𝑛 𝑋𝐴. 𝑋𝐵 = 𝑋𝐶. 𝑋𝐷
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3 Unit Maths – Circle Geometry
Intersecting segment theorem: if two secants intersect outside a circle, the product of one secant and its
external segment equals the product of the other and its external segment.
𝑖𝑓 𝑋𝐴𝐵 𝑎𝑛𝑑 𝑋𝐶𝐷 𝑎𝑟𝑒 𝑠𝑒𝑐𝑎𝑛𝑡𝑠 𝑡ℎ𝑒𝑛 𝑋𝐴. 𝑋𝐵 = 𝑋𝐶. 𝑋𝐷
If a tangent and a secant intersect, the product of the secant and its external segment equals the square of
the tangent.
𝐼𝑓 𝑋𝑇 𝑖𝑠 𝑎 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑎𝑛𝑑 𝑋𝐴𝐵 𝑖𝑠 𝑎 𝑠𝑒𝑐𝑎𝑛𝑡 𝑡ℎ𝑒𝑛 𝑋𝐴. 𝑋𝐵 = 𝑋𝑇 2
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