162 Complex coordinates; sensed angles; angles between lines Ch. 10
ProofWe use the notation of 10.6.1. In the case (10.6.1)z′=z 0 +zcisαso that
σ=
z′ 4 −z′ 3
z′ 2 −z′ 1
=
(z 4 −z 3 )cisα
(z 2 −z 1 )cisα
=ρ.
In the case (10.6.2)z′=z 0 + ̄zcisαso that
σ=
z′ 4 −z′ 3
z′ 2 −z′ 1
=
(z ̄ 4 −z ̄ 3 )cisα
(z ̄ 2 −z ̄ 1 )cisα
=ρ ̄.
10.8 Sensed angles of triangles, the sine rule ...............
10.8.1 .....................................
Definition. For any non-full angleθ, we denote byθFtheangleinA(F)such that
|θF|◦=|θ|◦.
O I
J H^1
H 2
H 4 H 3
Z 1
Z 2
Z 3
|β|◦
|α|◦
|γ|◦
a
c
b
Figure 10.7.
O I
J H^1
H 2
H 4 H 3
Z 1
Z 3
Z 2
|γ|◦
|α|◦
|β|◦
a
c
b
NOTATION. For non-collinear pointsZ 1 ,Z 2 ,Z 3 , we use as standard notation
|Z 2 ,Z 3 |=a,|Z 3 ,Z 1 |=b,|Z 1 ,Z 2 |=c,
u=
b
c
,v=
c
a
,w=
a
b
,
α=FZ 2 Z 1 Z 3 ,β=FZ 3 Z 2 Z 1 ,γ=FZ 1 Z 3 Z 2.
Note that by comparison with 9.5.1 we are now using sensed-angles instead of wedge-
angles.
