Geometry with Trigonometry

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

⎞

⎠= 0.