Geometry with Trigonometry

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

When this holds and Z and Z 2 are on the same side of Z 3 Z 4 ,then

(z−z 3 )(z 2 −z 4 )

(z−z 4 )(z 2 −z 3 )

> 0.

Proof. The given condition is equivalent to

z−z 4

z−z 3

=t

z 2 −z 4

z 2 −z 3

, (10.9.1)

for somet=0inR.LetG 1 ,G 2 be the open half-planes with common edgeZ 3 Z 4 , with
Z 2 ∈G 1 .Letθ=FZ 3 Z 2 Z 4 andφ=FZ 3 ZZ 4.
Suppose first that (10.9.1) holds. ForZ∈G 1 ,

ℑ

z−z 4

z−z 3

andℑ

z 2 −z 4

z 2 −z 3

must have the same sign and sot>0; it follows thatφ=θ.ForZ∈G 2 ,

ℑ

z−z 4

z−z 3

andℑ

z 2 −z 4

z 2 −z 3

must have opposite signs and sot<0; it follows thatφ=θ+ (^180) F′. By 10.9.2Z∈C
in both cases.
Conversely letZ∈C. Then by (10.9.1) forZ∈G 1 we haveφ=θ, while for
Z∈G 2 we haveφ=θ+ (^180) F′and the result now follows.
The expression((zz−−zz 43 )()(zz^22 −−zz 34 ))is called thecross-ratioof the ordered set of points
(Z,Z 2 ,Z 3 ,Z 4 ).

10.9.4 Ptolemy’s theorem, c. 200A.D. ....................

Let Z 2 ,Z 3 ,Z 4 be non-collinear
points and C the circle that
contains them. Let Z∈C be
such that Z and Z 3 are on
opposite sides of Z 2 Z 4 .Then
|Z,Z 4 ||Z 2 ,Z 3 |+|Z,Z 2 ||Z 3 ,Z 4 |=
|Z,Z 3 ||Z 2 ,Z 4 |.

Z

Z 2

Z 3

Z 4

W

Figure 10.10.

Proof. By multiplying out, it can be checked that

(z−z 4 )(z 2 −z 3 )+(z−z 2 )(z 3 −z 4 )=(z−z 3 )(z 2 −z 4 ).