Circle theorems There are five main theorems, which relate to or drawn inside the circumference of a circle.

ac180o bd180o

‘Arrowhead’ ‘Right- diameter’ ‘Mountain’ or ‘bow-tie’ theorem ‘Cyclic ’ theorem Chord-tangent or Alternate segment theorem theorem theorem The in the same If a 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 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 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: – 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.

The obtuse angle AOB = 2a is the same for both cb90o 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. cb90o 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 angle at AOC. i.e. reflex angles as ac180o b d  e  360o well as obtuse or indeed acute o o bd180 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