Circle theorems       There are five main circle theorems, which relate to triangles or quadrilaterals drawn inside the circumference of a circle. 
                                                                                                                                             ac180o
                                                                                                                                             bd180o
              ‘Arrowhead’                             ‘Right-angle diameter’               ‘Mountain’ or ‘bow-tie’ theorem              ‘Cyclic quadrilateral’ theorem            Chord-tangent or Alternate segment theorem 
              theorem                                 theorem 
      An angle at the centre of a circle is           Any angle (inscribed)                 The angles in the same                   A quadrilateral ABCD is cyclic if            If a line drawn through the end point of a chord 
      twice (the size of) the angle on the            in a semicircle is a                  segment (subtended                       and only if (it is convex and )              forms an angle equal to the angle subtended by 
      circumference if they are both                  right angle.                          by the same arc or                       both pairs of opposite angles                the chord in the alternate segment then the line 
      subtended by the same arc.                                                            arcs of the same size)                   are supplementary                            is a tangent (chord-tangent or alternate 
                                                                                            are equal.                                                                            segment theorem) 
         Internal angles of any triangle sum to 180o                                                                             arc               Proof of ‘Right-angle diameter’ 
                                                                                                                                                   theorem 
                                                                                                                  radius                            
                                                                                                                                                   This is a special case of the 
                                                                                                        chord                                      ‘Arrowhead’ theorem: 
                                              ABC180o                                                                                            
                                                                                      segment                                                                         o
                                                                                                                                                   When 2x = 180   
                                                                                                                        sector                     this means the arrowhead angle                                           ‘Arrowhead’
                                                                                                                                                                                o                                           theorem 
                                                                                                                                                   x is half this, i.e. x = 90 . 
        Proof of the ‘Arrowhead’ theorem 
                                                   2ad180o
                                                                  o    Add these together ... 
                                                   2bc180 
                                                  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                                                               o
                                                                                  ‘Arrowhead’                                                                          2ac2    180
      Consider two arrowheads drawn from the same                                 theorem                                                                              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  cb90o
      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 
                                                                          bde360o                  sequence, BD is a 
                                                                                                      constant, but the chord 
                                                                          2ab180o                   BC tends to zero. 
                                                                          2cd180o
                                                                          2(ac)bd bde
                                                                          2(ac)e             Which essentially shows the 
                                                                                                ‘Arrowhead’ theorem 
     From the ‘Arrowhead’ theorem                                                               generalizes for any ‘external’ 
     2f b d                                                                                   angle at AOC. i.e. reflex angles as           ac180o
                                       bde360o
                                                                                                well as obtuse or indeed acute                bd180o
                  Putting these        2f 2(ac)360o                                         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 
        There are two other circle theorems in addition to the main five                                            Secant / Tangent theorem        ACBA AD2
        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 
           AXBX CXDX                                                                                                                                Use the Alternate segment 
                                                                                                                                                       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 
          between corresponding sides must be the same                BX    DX                                          between corresponding sides must be the same 
                                                                 k 
                                                                     CX     AX                                              AD AC
                                                                                                                        k 
                                                                  AXBX CXDX                                             BA     AD
                                                                                                                        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,   2                              2      2
                                                                                                                                                    h r         b
                                                                                           the tangent lengths CB and AB must be  
                                                                                           the same. 
                                                                                                     
                                                                                                     
                                                                                                     
                                                                                                                                                                    Mathematics topic handout: Geometry – Circle theorems  Dr Andrew French. www.eclecticon.info  PAGE 4