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Circle Theorems There Are Five Main Circle Theorems, Which Relate to Triangles Or Quadrilaterals Drawn Inside the Circumference of a Circle

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Circle Theorems There Are Five Main Circle Theorems, Which Relate to Triangles Or Quadrilaterals Drawn Inside the Circumference of a Circle Circle theorems There are five main circle theorems, which relate to triangles or quadrilaterals drawn inside the circumference of a circle. ac180o bd180o ‘Arrowhead’ ‘Right-angle diameter’ ‘Mountain’ or ‘bow-tie’ theorem ‘Cyclic quadrilateral’ theorem Chord-tangent or Alternate segment theorem theorem theorem The angles in the same If a line drawn through the end point of a chord An angle at the centre of a circle is Any angle (inscribed) A quadrilateral ABCD is cyclic if segment (subtended forms an angle equal to the angle subtended by twice (the size of) the angle on the in a semicircle is a and only if (it is convex and ) by the same arc or the chord in the alternate segment then the line circumference if they are both both pairs of opposite angles right angle. arcs of the same size) is a tangent (chord-tangent or alternate subtended by the same arc. are supplementary are equal. segment theorem) Proof of ‘Right-angle diameter’ Internal angles of any triangle sum to 180o arc theorem radius This is a special case of the chord o ‘Arrowhead’ theorem: ABC 180 segment When 2x = 180o sector this means the arrowhead angle ‘Arrowhead’ x is half this, i.e. x = 90o. theorem Proof of the ‘Arrowhead’ theorem 2ad 180o Add these together ... 2bc 180o 2(a b ) d c 360o d c e 360o d c e 2( a b ) d c e 2( a b ) These are isosceles triangles since they both meet at the origin of the circle, and therefore two edges of each triangle are circle radii. Mathematics topic handout: Geometry – Circle theorems Dr Andrew French. www.eclecticon.info PAGE 1 Proof of the Alternate From the diagram Proof of the ‘Mountain’ theorem segment theorem ‘Arrowhead’ 2ac 2 180o theorem Consider two arrowheads drawn from the same ac 90o points A and B on the circle perimeter. ac180o o The obtuse angle AOB = 2a is the same for both bd180 cb90o arrowheads. c b a c By the ‘Arrowhead’ theorem, the arrowhead angle must be half this, i.e. a. ba Hence the arrowhead angles at C and C’ must both be a. cb90o The ‘Mountain’ theorem is so named because the angles at C and C’ look a little like the snowy Note DE is a peaks of mountains! tangent to the circle at point A The ‘Searchlight’, or ‘bow-tie’ theorem is hence another popular name, for similar visual reasons. This can be proven by application of the ‘right Proof of the ‘Cyclic quadrilateral’ theorem angle diameter’ theorem . In the picture o sequence, BD is a b d e 360 constant, but the chord 2ab 180o BC tends to zero. 2cd 180o 2(a c ) b d b d e Which essentially shows the 2(a c ) e ‘Arrowhead’ theorem From the ‘Arrowhead’ theorem generalizes for any ‘external’ 2 f b d o angle at AOC. i.e. reflex angles as b d e 360 well as obtuse or indeed acute o Putting these 2f 2( a c ) 360 varieties. results together f a c 180o i.e. the opposite angles of a cyclic quadrilateral sum to 180o Mathematics topic handout: Geometry – Circle theorems Dr Andrew French. www.eclecticon.info PAGE 2 2 There are two other circle theorems in addition to the main five Secant / Tangent theorem AC BA AD Intersecting chords theorem Firstly label internal angles a, b, c One can easily prove this result using the ‘Mountain Theorem’ to label the internal angles Use the Alternate segment AX BX CX DX theorem to show that angle ADB is also c Hence angle ADC is b Triangles ACX and DBX are therefore similar Triangles ABD and ADC are therefore similar enlargement by k enlargement by k Hence the enlargement factor k Hence the enlargement factor k BX DX between corresponding sides must be the same k between corresponding sides must be the same CX AX AD AC k BA AD AX BX CX DX AC BA AD2 Mathematics topic handout: Geometry – Circle theorems Dr Andrew French. www.eclecticon.info PAGE 3 Further circle theorem notes Tangents from an external point are equal in length. This is perhaps obvious on symmetry grounds, but can be proven formally since triangles OCB and OAB have the following properties: (i) A right angle at, respectively, A and C since lines AB and CB are tangents to the circle (ii) The sides OC and OA are circle radii so must be the same length (iii) The side OB is common to both triangles Hence using Pythagoras’ Theorem, h2 r 2 b 2 the tangent lengths CB and AB must be the same. Mathematics topic handout: Geometry – Circle theorems Dr Andrew French. www.eclecticon.info PAGE 4 .
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