164 Complex coordinates; sensed angles; angles between lines Ch. 10
10.9 Someresultsoncircles ........................
10.9.1 A necessary condition to lie on a circle ...............
In this section we provide some results on circles which are conveniently proved
using complex coordinates.
Let Z 1 ,Z 2 be fixed distinct
points, and Z a variable point,
all on the circle C(Z 0 ;k).
Let F′ = tO,Z 0 (F) and
α=∠F′I 0 Z 0 Z 1 ,β=∠F′I 0 Z 0 Z 2
andγ=^12 (β−α). As Z varies
on the circle, in one of the open
half-planes with edge Z 1 Z 2
the sensed angle FZ 1 ZZ 2 is
equal in measure to γ, while
in the other open half-plane
with edge Z 1 Z 2 it is equal in
measure toγ+ (^180) F′. Note that
2 γ=FZ 1 Z 0 Z 2.
O I
J H^1
H 2
H 4 H 3
Z 0 I 0
J 0
Z 1
Z 2
Z
α
β
Figure 10.8.
Proof.Nowz 1 −z 0 =kcisα,z 2 −z 0 =kcisβand ifθ=FI 0 Z 0 Z,thenz−z 0 =
kcisθ.We writeφ=FZ 1 ZZ 2 so that
z 2 −z
z 1 −z
=lcisφ,wherel=
|Z,Z 2 |
|Z,Z 1 |
.
Then
lcisφ=
cisβ−cisθ
cisα−cisθ
,
while on taking complex conjugates here
lcis(−φ)=
cis(−β)−cis(−θ)
cis(−α)−cis(−θ)
=
cisα
cisβ
cisθ−cisβ
cisθ−cisα
.
By division
cis 2φ=
cisβ
cisα
=cis(β−α).
Thus 2(cisφ)^2 =(cisγ)^2 so that cisφ=±cisγ. Thus either cisφ=cisγor cisφ=
cis(γ+ (^180) F′),and accordingly
ℑ
z 2 −z
z 1 −z
=lsinγ orℑ
z 2 −z
z 1 −z
=lsin(γ+ (^180) F′).
As sinγ> 0 ,the first of these occurs whenZis in the half-plane with edgeZ 1 Z 2 in
whichℑzz^21 −−zz> 0 ,and the second whenZis in the half-plane with edgeZ 1 Z 2 in which
ℑzz 12 −−zz< 0.