Sec. 11.6 Mobile coordinates 213
The mid-point sought is
z 3 +
1
2 |Z 2 ,Z 3 |
⎡
⎣z 2 −z 3 +√^1 −p^1 −ıq^1
( 1 −p 1 )^2 +q^21
(z 2 −z 3 )
⎤
⎦
=z 3 +
1
2 |Z 2 ,Z 3 |
⎡
⎣ 1 +√^1 −p^1 −ıq^1
( 1 −p 1 )^2 +q^21
⎤
⎦
Then points on the midline of|Z 2 Z 3 Z 1 have complex coordinates of the form
z 3 + s
2 |Z 2 ,Z 3 |
⎛
⎝√^1 −p^1 +q^1 ı
(p 1 − 1 )^2 +q^21
⎞
⎠(z 2 −z 3 )
for real numberss.
For a point of intersection of the two mid-lines we need
z 2 +
r
2 |Z 2 ,Z 3 |
⎛
⎝ 1 +√p^1 +q^1 ı
(p^21 +q^21
⎞
⎠(z 3 −z 2 )
=z 3 + s
2 |Z 2 ,Z 3 |
⎛
⎝ 1 +√^1 −p^1 −q^1 ı
(p 1 − 1 )^2 +q^21
⎞
⎠(z 2 −z 3 )
=z 3 −
s
2 |Z 2 ,Z 3 |
⎛
⎝ 1 +√^1 −p^1 −q^1 ı
(p 1 − 1 )^2 +q^21
⎞
⎠(z 3 −z 2 ).
Then
z 3 −z 2 +
+
⎡
⎣ −r
2 |Z 2 ,Z 3 |
( 1 +
p 1 +ıq 1
√
p^21 +q^21
−
s
2 |Z 2 ,Z 3 |
( 1 +
1 −p 1 −ıq 1
√
( 1 −p 1 )^2 +q^21
)
⎤
⎦(z 3 −z 2 )= 0 ,
1 +
−r
2 |Z 2 ,Z 3 |
( 1 +
p 1 +ıq 1
√
p^21 +q^21
)−
s
2 |Z 2 ,Z 3 |
( 1 +
1 −p 1 −ıq 1
√
( 1 −p 1 )^2 +q^21
)= 0.
Equating to 0 the imaginary part we obtain
−
rq 1
√
p^21 +q^21
+
sq 1
√
( 1 −p 1 )^2 +q^21
= 0 ,
and the real part
1 −
r
2 |Z 2 ,Z 3 |
⎛
⎝ 1 +√ p^1
p^21 +q^21
⎞
⎠− s
2 |Z 2 ,Z 3 |
⎛
⎝ 1 +√^1 −p^1
( 1 −p 1 )^2 +q^21