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Transcript 
3 Unit Maths – Circle Geometry
Terms
Chord
A chord is a line segment joining any two points on the circle.
ο‚· 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 βˆ π΄π‘‚π΅ = βˆ πΆπ‘‚π· π‘‘β„Žπ‘’π‘› π‘Žπ‘Ÿπ‘ 𝐴𝐡 = π‘Žπ‘Ÿπ‘ 𝐢𝐷
𝑖𝑓 π‘Žπ‘Ÿπ‘ 𝐴𝐡 = π‘Žπ‘Ÿπ‘ 𝐢𝐷 π‘‘β„Žπ‘’π‘› βˆ π΄π‘‚π΅ = βˆ πΆπ‘‚π·
Equal chords subtend equal arcs, and vice versa
𝑖𝑓 𝐴𝐡 = 𝐢𝐷 π‘‘β„Žπ‘’π‘› π‘Žπ‘Ÿπ‘ 𝐴𝐡 = π‘Žπ‘Ÿπ‘ 𝐢𝐷
𝑖𝑓 π‘Žπ‘Ÿπ‘ 𝐴𝐡 = π‘Žπ‘Ÿπ‘ 𝐢𝐷 π‘‘β„Žπ‘’π‘› 𝐴𝐡 = 𝐢𝐷
Equal chords are equidistant from the centre, and vice versa
𝐸 π‘Žπ‘›π‘‘ 𝐹 π‘Žπ‘Ÿπ‘’ π‘‘β„Žπ‘’ π‘šπ‘–π‘‘π‘π‘œπ‘–π‘›π‘‘π‘  π‘œπ‘“ π‘‘β„Žπ‘’ π‘β„Žπ‘œπ‘Ÿπ‘‘π‘  𝐴𝐡 π‘Žπ‘›π‘‘ 𝐢𝐷 π‘Ÿπ‘’π‘ π‘π‘’π‘π‘‘π‘–π‘£π‘’π‘™π‘¦
𝑖𝑓 𝐴𝐡 = 𝐢𝐷 π‘‘β„Žπ‘’π‘› 𝑂𝐸 = 𝑂𝐹
𝑖𝑓 π‘π‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ π‘π‘–π‘ π‘’π‘π‘‘π‘œπ‘Ÿπ‘  𝑂𝐸 = 𝑂𝐹 π‘‘β„Žπ‘’π‘› 𝐴𝐡 = 𝐢𝐷
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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 π‘‘β„Žπ‘’ π‘šπ‘–π‘‘π‘π‘œπ‘–π‘›π‘‘ π‘œπ‘“ 𝐴𝐡 π‘Žπ‘›π‘‘ 𝑂 𝑖𝑠 π‘‘β„Žπ‘’ π‘π‘’π‘›π‘‘π‘Ÿπ‘’, π‘‘β„Žπ‘’π‘› 𝑂𝐸 βŠ₯ 𝐴𝐡
𝑖𝑓 𝑂𝐸 βŠ₯ 𝐴𝐡, π‘€β„Žπ‘’π‘Ÿπ‘’ 𝑂 𝑖𝑠 π‘‘β„Žπ‘’ π‘π‘’π‘›π‘‘π‘Ÿπ‘’ π‘‘β„Žπ‘’π‘› 𝐸 𝑖𝑠 π‘‘β„Žπ‘’ π‘šπ‘–π‘‘π‘π‘œπ‘–π‘›π‘‘ π‘œπ‘“ 𝐴𝐡
𝑖𝑓 𝑂𝐸 βŠ₯ 𝐴𝐡, π‘€β„Žπ‘’π‘Ÿπ‘’ 𝐸 𝑠 π‘‘β„Žπ‘’ π‘šπ‘–π‘‘π‘π‘œπ‘–π‘›π‘‘ π‘œπ‘“ 𝐴𝐡 π‘‘β„Žπ‘’π‘› 𝑂 𝑖𝑠 π‘‘β„Žπ‘’ π‘π‘’π‘›π‘‘π‘Ÿπ‘’
The angle at the centre is twice the angle at the circumference subtended by the same arc
βˆ π΄π‘‚π΅ = 2βˆ π΄π‘ƒπ΅
Angles subtending the same arc are equal
𝑖𝑓 𝑃, 𝑄, 𝐴, 𝐡 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘›π‘π‘¦π‘π‘™π‘–π‘ π‘‘β„Žπ‘’π‘› βˆ π΄π‘ƒπ΅ = βˆ π΄π‘„π΅
𝑖𝑓 βˆ π΄π‘ƒπ΅ = βˆ π΄π‘„π΅ π‘‘β„Žπ‘’π‘› 𝑃, 𝑄, 𝐴, 𝐡 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘›π‘π‘¦π‘π‘™π‘–π‘
The angle on a semi-circle is a right angle, and vice versa
𝑖𝑓 𝐴𝐡 𝑖𝑠 π‘Ž π‘‘π‘–π‘Žπ‘šπ‘’π‘‘π‘’π‘Ÿ π‘‘β„Žπ‘’π‘› βˆ π΄π‘ƒπ΅ = 90°
𝑖𝑓 βˆ π΄π‘ƒπ΅ = 90° π‘‘β„Žπ‘’π‘› 𝐴𝐡 𝑖𝑠 π‘Ž π‘‘π‘–π‘Žπ‘šπ‘’π‘‘π‘’π‘Ÿ
The opposite angles of a cyclic quadrilateral are supplementary, and vice versa
𝑖𝑓 𝐴, 𝐡, 𝐢, 𝐷 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘›π‘π‘¦π‘π‘™π‘–π‘ π‘‘β„Žπ‘’π‘› ∠𝐴𝐡𝐢 + ∠𝐴𝐷𝐢 = 180°
𝑖𝑓 ∠𝐴𝐡𝐢 + ∠𝐴𝐷𝐢 = 180° π‘‘β„Žπ‘’π‘› 𝐴, 𝐡, 𝐢, 𝐷 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘›π‘π‘¦π‘π‘™π‘–π‘
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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, 𝐡, 𝐢, 𝐷 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘›π‘π‘¦π‘π‘™π‘–π‘ π‘‘β„Žπ‘’π‘› ∠𝐴𝐡𝐢 = βˆ πΆπ·π‘‹
𝑖𝑓 ∠𝐴𝐡𝐢 = βˆ πΆπ·π‘‹ π‘‘β„Žπ‘’π‘› 𝐴, 𝐡, 𝐢, 𝐷 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘›π‘π‘¦π‘π‘™π‘–π‘
The tangent to a circle is perpendicular to the radius drawn to the point of contact, and vice versa
𝑇 𝑖𝑠 π‘‘β„Žπ‘’ π‘π‘œπ‘–π‘›π‘‘ π‘œπ‘“ π‘π‘œπ‘›π‘‘π‘Žπ‘π‘‘ π‘œπ‘“ π‘‹π‘Œ π‘Žπ‘›π‘‘ π‘‘β„Žπ‘’ π‘π‘–π‘Ÿπ‘π‘™π‘’
𝑖𝑓 𝑂𝑇 βŠ₯ π‘‹π‘Œ π‘‘β„Žπ‘’π‘› π‘‹π‘Œ 𝑖𝑠 π‘Ž π‘‘π‘Žπ‘›π‘”π‘’π‘›π‘‘
𝑖𝑓 π‘‹π‘Œ 𝑖𝑠 π‘Ž π‘‘π‘Žπ‘›π‘”π‘’π‘›π‘‘ π‘‘β„Žπ‘’π‘› 𝑂𝑇 βŠ₯ π‘‹π‘Œ
Tangents to a circle from an external point are equal
𝑖𝑓 𝑃𝐴 π‘Žπ‘›π‘‘ 𝑃𝐡 π‘Žπ‘Ÿπ‘’ π‘‘π‘Žπ‘›π‘”π‘’π‘›π‘‘π‘  π‘‘β„Žπ‘’π‘› 𝑃𝐴 = 𝑃𝐡
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 π‘‹π‘Œ 𝑖𝑠 π‘Ž π‘‘π‘Žπ‘›π‘”π‘’π‘›π‘‘ π‘‘β„Žπ‘Žπ‘‘ π‘‘π‘œπ‘’π‘β„Žπ‘’π‘  π‘‘β„Žπ‘’ π‘π‘–π‘Ÿπ‘π‘™π‘’ 𝑖𝑛 𝐴 π‘‘β„Žπ‘’π‘› βˆ π΄π‘ƒπ΅ = βˆ π‘‹π΄π΅
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 π‘‘π‘€π‘œ π‘β„Žπ‘œπ‘Ÿπ‘‘π‘  𝐴𝐡 π‘Žπ‘›π‘‘ 𝐢𝐷 π‘–π‘›π‘‘π‘’π‘Ÿπ‘ π‘’π‘π‘‘ π‘Žπ‘‘ 𝑋 π‘‘β„Žπ‘’π‘› 𝑋𝐴. 𝑋𝐡 = 𝑋𝐢. 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 𝑋𝐴𝐡 π‘Žπ‘›π‘‘ 𝑋𝐢𝐷 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘Žπ‘›π‘‘π‘  π‘‘β„Žπ‘’π‘› 𝑋𝐴. 𝑋𝐡 = 𝑋𝐢. 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 π‘Ž π‘‘π‘Žπ‘›π‘”π‘’π‘›π‘‘ π‘Žπ‘›π‘‘ 𝑋𝐴𝐡 𝑖𝑠 π‘Ž π‘ π‘’π‘π‘Žπ‘›π‘‘ π‘‘β„Žπ‘’π‘› 𝑋𝐴. 𝑋𝐡 = 𝑋𝑇 2
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