224 Vector and complex-number methods Ch. 11
Then
p^24 +q^24 =p^24 +(
(p 1 − 1 )
q 1
)^2 p^24 ,
p^24 +q^24
q 4
=
p^24
q 4
(
1 +(
p 1 − 1
q 1
)^2
)
=p 4.
−q 1
p 1 − 1
(
1 +(
p 1 − 1
q 1
)^2
)
=
2 p 1 − 1
p 1
1
1 +(p^1 q− 11 )^2
−q 1
p 1 − 1
(
1 +(
p 1 − 1
q 1
)^2
)
=−
2 p 1 − 1
p 1
q 1
p 1 − 1
.
By (11.6.4) the circumcentreZ( 16 iii)of[Z 4 ,Z 8 ,Z 18 ]where
z 4 =z 8 +(p 4 +q 4 ı)(z 18 −z 8 )
is given by
z( 16 iii)=z 8 +^12
[
1 +
p^24 +q^24
q 4
ı−
p 4
q 4
ı
]
(z 18 −z 8 )
=z 8 +^12
⎡
⎣ 1 +
p^24 ( 1 +(p^1 −^1 )
2
q^21 )
q 4
ı+
q 1
p 1 − 1
ı
⎤
⎦(z 18 −z 8 )
=z 8 +^12
[
1 +p 4
−q 1
p 1 − 1
(
1 +
(p 1 − 1 )^2
q^21
)
ı+
q 1
p 1 − 1
ı
]
(z 18 −z 8 )
=z 8 +^12
[
1 −
2 p 1 − 1
p 1
q 1
p 1 − 1
ı+
q 1
p 1 − 1
ı+
q 1
p 1 − 1
ı
]
(z 18 −z 8 )
=z 8 +^12
[
1 +
q 1
p 1 − 1
(
−
2 p 1 − 1
p 1
+ 1
)
ı
]
(z 24 −z 8 )
=z 8 +^12 [ 1 −
q 1
p 1
ı](z 18 −z 8 ).
Continuing from this we have