212 Vector and complex-number methods Ch. 11
Similarly the point with complex coordinatez 2 +|Z 21 ,Z 1 |(z 1 −z 2 ), lies on the half-line
[Z 2 ,Z 1 at a unit distance fromZ 2. We wish to consider the mid-point of these points
with complex coordinates
z 2 +
1
|Z 2 ,Z 3 |
(z 3 −z 2 ),z 2 +
1
|Z 2 ,Z 1 |
(z 1 −z 2 ).
Nowz 1 −z 2 =(p 1 +ıq 1 )(z 3 −z 2 )so that
|z 1 −z 2 |=|p 1 +ıq 1 ||z 3 −z 2 |=
√
p^21 +q^21 |z 3 −z 2 |.
Then
z 1 −z 2
|Z 2 ,Z 1 |
=
p 1 +ıq 1
√
p^21 +q^21
z 3 −z 2
|Z 2 ,Z 3 |
.
The mid-point of these two points has complex coordinate
z 2 +
1
2 |Z 2 ,Z 3 |
⎡
⎣z 3 −z 2 +√^1
p^21 +q^21
(z 1 −z 2 )
⎤
⎦
=z 2 +
1
2 |Z 2 ,Z 3 |
⎡
⎣z 3 −z 2 +√p^1 +q^1 ı
p^21 +q^21
(z 3 −z 2 )
⎤
⎦
=z 2 +^1
2 |Z 2 ,Z 3 |
⎡
⎣ 1 +√p^1 +ıq^1
p^21 +q^21
⎤
⎦(z 3 −z 2 ).
Then points on the midline of|Z 1 Z 2 Z 3 then have complex coordinates of the form
z 2 +
r
2 |Z 2 ,Z 3 |
⎡
⎣ 1 +√p^1 +q^1 ı
(p^21 +q^21
⎤
⎦(z 3 −z 2 ),
for real numbersr.
By a similar argument the point with complex coordinatez 3 +|Z 21 ,Z 3 |(z 2 −z 3 )lies
on the half-line[Z 3 ,Z 2 at a unit distance fromZ 3. Also the point with complex coor-
dinatez 3 +|Z 31 ,Z 1 |(z 1 −z 3 ), lies on the half-line[Z 3 ,Z 1 at a unit distance fromZ 3 .We
wish to consider the mid-point of these points with complex coordinates
z 3 +
1
|Z 2 ,Z 3 |
(z 2 −z 3 ), z 3 +
1
|Z 3 ,Z 1 |
(z 1 −z 3 ).
As preparation we note that
z 1 =z 2 +(p 1 +ıq 1 )(z 3 −z 2 )=z 3 +(p 1 − 1 +ıq 1 )(z 3 −z 2 )
z 1 −z 3 =( 1 −p 1 −ıq 1 )(z 2 −z 3 ), |Z 1 ,Z 3 |=
√
( 1 −p 1 )^2 +q^21 |Z 2 ,Z 3 |,
z 1 −z 3
|Z 3 ,Z 1 |
=
1 −p 1 −ıq 1
[
√
( 1 −p 1 )^2 +q^21
z 2 −z 3
|Z 2 ,Z 3 |