history of mathematics - National STEM Centre

"fJotice that Figure 4.5 is
not drawn to scale.
Figure 4.5
4 More Greek mathematics
In Figure 4.5, NC = sn and the chord AC with length 2sn is at a distance cn from
the centre of the circle. In the triangle OBC, the chord BC from the next iteration
has length 2sn+t , and its distance from the centre is cn+] . The tangent at B has
length 2tn , while the tangent at C, from the next iteration, has total length 2tn+t .
a Show that CD = 2cn+1 .
b By calculating the area of the triangle BCD in two different ways, show that
s= Sc '
c Use the similar triangles CDN and BDC to show that —— -
thatcn+1 =^-^=-.
d Use similar triangles to show that tn = —
, and hence
Let Sn be half the perimeter of the inside 6x2" -sided polygon after n iterations, and
Tn be the corresponding semi-perimeter of the corresponding outside polygon.
3 a Show that Sn =3x2 n+[ sn andTn = 3x2 n+l tn .
b Use the results of part a and of question 2 to show that
C _Ty9" + 2 c -
^n+l - JX z sn+l ~
— ^ v 9"
— J A ^.
cn+\
3x2" +2 .
4 Use the three recurrence relations
_ sa
and
and
47