Sec. 10.9 Some results on circles 165
10.9.2 A sufficient condition to lie on a circle ...............
Let Z 1 ,Z 2 be fixed distinct points
and Z a variable point. As Z
varies in one of the half-planes
with edge Z 1 Z 2 , for the sensed
angleθ=FZ 1 ZZ 2 let|θ|◦=
|γ|◦ where γ is a fixed non-
null and non-straight angle in
A(F′), while as Z varies in the
other half-plane with edge Z 1 Z 2 ,
let|θ|◦=|γ+ (^180) F′|◦.ThenZ
lies on a circle which passes
through Z 1 and Z 2.
O I
J H^1
H 2
H 4 H 3
Z 0 I 0
J 0
Z 1
Z 2
Z
Figure 10.9.
Proof.Wehave
z 2 −z
z 1 −z
=tcisγ
for somet∈R{ 0 }.Then
z=
z 2 −tz 1 cisγ
1 −tcisγ
so that with cotγ=cosγ/sinγ,
z−
1
2
(z 1 +z 2 )−
1
2
ıcotγ.(z 2 −z 1 )
=
z 2 −tz 1 cisγ
1 −tcisγ
−
1
2
(z 1 +z 2 )−
1
2
icotγ.(z 2 −z 1 )
=
1
2 (z^2 −z^1 )[^1 +tcisγ−ıcotγ(^1 −tcisγ)]
1 −tcisγ
=
1
2 (z^2 −z^1 )[sinγ(^1 +tcisγ)−ıcosγ(^1 −tcisγ)]
sinγ( 1 −tcisγ)
=
1
2 (z^2 −z^1 )[sinγ+ı(t−cosγ)]
sinγ( 1 −tcisγ)
and this has absolute value
|z 2 −z 1 |
2 |sinγ|
This shows thatZlies on a circle, the centre and length of radius of which are evident.
10.9.3Complexcross-ratio ..........................
Let Z 2 ,Z 3 ,Z 4 be non-collinear points andCthe circle that contains them. Then Z∈
Z 3 Z 4 lies inCif and only if
ℑ
(z−z 3 )(z 2 −z 4 )
(z−z 4 )(z 2 −z 3 )