Geometry with Trigonometry

154 Complex coordinates; sensed angles; angles between lines Ch. 10

(iii) If the points Z 1 and Z 2 are both distinct from Z 0 ,then

FZ 1 Z 0 Z 2 =−FZ 2 Z 0 Z 1.

(iv) If Z 1 ,Z 2 ,Z 3 are all distinct from Z 0 ,then

FZ 1 Z 0 Z 2 +FZ 2 Z 0 Z 3 =FZ 1 Z 0 Z 3.

(v)Ifφ=FZ 1 Z 0 Z 2 ,thenrφ;Z 0 ([Z 0 ,Z 1 )=[Z 0 ,Z 2.

(vi)In10.3.1(ii),FZZ 0 Z′=α.

O I

J H^1

H 2

H 4 H 3

Z 0 I 0

J 0

Z 1

Z 2

θ 1

θ 2

φ

O I

J H^1

H 2

H 4 H 3

Z 0 I 0

J 0

Z 1

Z 2

θ 1

θ 2

θ 2 −θ 1

Z 3

θ 3

θ 3 −θ 2

Figure 10.5.

Proof.(i) Forz 1 −z 0 =|z 1 −z 0 |cisθ 1 ,z 2 −z 0 =|z 2 −z 0 |cisθ 2 and so

z 2 −z 0

z 1 −z 0

=

|z 2 −z 0 |cisθ 2

|z 1 −z 0 |cisθ 1

=

|z 2 −z 0 |

|z 1 −z 0 |

cis(θ 2 −θ 1 ).