Geometry with Trigonometry

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 |

.