230 Vector and complex-number methods Ch. 11
s(z 1 −z 3 ).Butz 1 −z 3 =(p 1 − 1 +q 1 ı)(z 3 −z 2 ),soz 3 −z 2 =p 1 − 11 +q 1 ı(z 1 −z 3 ).On
inserting this, we have thatz−w 2 =
[
p− 1 +qı
p 1 − 1 +q 1 ı−s]
(z 1 −z 3 ). We wish the coefficientofz 1 −z 3 to be purely imaginary, and so takes=(p−(^1 p)( 1 −p^11 −) (^21) +)+q 2 qq^1
1
. Hencew 2 =z 3 +
(p− 1 )(p 1 − 1 )+qq 1
(p 1 − 1 )^2 +q^21 (z^1 −z^3 ).
Thirdly,w 3 =z 2 +t(z 1 −z 2 ),for somet∈R. Hencez−w 3 =(p+qı)(z 3 −z 2 )−
t(z 1 −z 2 ).Butz 3 −z 2 =p 1 +^1 q 1 ı(z 1 −z 2 )and soz−w 3 =
[p+qı
p 1 +q 1 ı−t]
(z 1 −z 2 ).Wechoosetso that the coefficient ofz 1 −z 2 is purely imaginary. Thust=ppp 112 ++qqq 211 which
yieldsw 3 =z 2 +ppp^12 +qq^1
1 +q^21
(z 1 −z 2 ).
From these expressions forw 1 ,w 2 ,w 3 we note thatw 2 −w 1
w 3 −w 1=
z 3 −z 2 +(p−(p^11 )(−p 11 )− (^2) +^1 )+q 2 qq^1
1
(z 1 −z 3 )−p(z 3 −z 2 )
pp 1 +qq 1
p^21 +q^21 (z^1 −z^2 )−p(z^3 −z^2 )
=
1 −p+(p−(^1 p)( 1 −p 11 )− (^21) +)+q 2 qq^1
1
(p 1 − 1 +q 1 ı)
pp 1 +qq 1
p^21 +q^21 (p^1 +q^1 ı)−p
.
The real part of this has numerator(p^21 +q^21 −p 1 )(p^2 +q^2 )−(p^21 +q^21 −p 1 )p+q 1 q,
and the imaginary part has numeratorq 1 (p^2 +q^2 )−q 1 p−(p^21 +q^21 −p 1 )q.Ifθ=
F′W 3 W 1 W 2 then forθto have a constant magnitude it is necessary and sufficient
that
q 1 (p^2 +q^2 )−q 1 p−(p^21 +q^21 −p 1 )q
=k[(p^21 +q^21 −p 1 )(p^2 +q^2 )−(p^21 +q^21 −p 1 )p+q 1 q].This can be re-written as
a^2[(
p−1
2
) 2
+
(
q−p^21 +q^21 −p 1
2 q 1) 2 ]
−
a^2
4[
1 +
(p^21 +q^21 −p 1 )^2
q^21]
=
k(p^21 +q^21 −p 1 )
q 1{
a^2[(
p−1
2
) 2
+
(
q+q 1
2 (p^21 +q^21 −p 1 )) 2 ]
−
a^2
4[
1 +
q^21
(p^21 +q^21 −p 1 )^2]}
.
On using 11.6.1 we infer that askvariesthis gives the family of coaxal circles which
pass through Z 2 and Z 3.
For W 1 ,W 2 ,W 3 to be collinear, it is necessary and sufficient that the expression
be real. On equating its imaginary part to 0 we obtainq 1 (p^2 +q^2 )−q 1 p−(p^21 +
q^21 −p 1 )q=0. On writing this as p
(^2) +q (^2) −p
q =
p^21 +q^21 −p 1
q 1 , we note from the formula
for a circumcentre in 11.6.4? that it holds whenZ lies on the circumcircle of the
triangle[Z 1 ,Z 2 ,Z 3 ].
This latter result is due toWallace,butSimson’sname has for a long time been
associated with it.