Geometry with Trigonometry

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

the terminology used being ‘cross’. Sensed duo-angles were also used by Johnson [9,
pages 11 –15] under the name of ‘directed angles’.
We have
tan(βd−αd)=

tanβd−tanαd

1 +tanαdtanβd

,

provided none ofαd,βd,βd−αdis a right duo-angle. For a coordinate formula to
utilise this we replacex 4 by−x 4 in (10.10.3) and translate to parallel axes through
Z 1. Thus forγd=F(Z 1 Z 4 ,Z 1 Z 5 )we have

tanγd=

y 5 −y 1

x 5 −x 1 −

y 4 −y 1

x 4 −x 1

1 +yx^55 −−yx^11 yx^44 −−yx^11

whenγdis not right, and

1 +

y 5 −y 1

x 5 −x 1

y 4 −y 1

x 4 −x 1

= 0

when it is.

10.10.9Anapplication ............................

For fixed pointsZ 4 andZ 5 , consider the locus of pointsZsuch that
F(ZZ 4 ,ZZ 5 )has constant magnitude. If it is a right duo-angle we will have

1 +

y 5 −y

x 5 −x

y 4 −y

x 4 −x

= 0 ,

and so the pointsZ∈Z 4 Z 5 lie on the circle on[Z 4 ,Z 5 ]as diameter. Otherwise, we
have that y 5 −y
x 5 −x−

y 4 −y

x 4 −x

1 +yx^55 −−xyyx^44 −−yx

= 1 −λ,

for someλ=1, and then the pointsZ∈Z 4 Z 5 lie on a circle which passes throughZ 4
andZ 5. In fact we obtain a set of coaxal circles throughZ 4 andZ 5. This should be
compared with 7.5.1 and 10.9.1.

10.11 A case of Pascal’s theorem, 1640 ...................

10.11.1 ....................................

Let Z 1 ,W 1 ,Z 2 ,W 2 be distinct points on the circleC(O;k).ThenZ 1 W 2 ‖W 1 Z 2 if and
only ifFZ 2 OW 2 =FZ 1 OW 1.
Proof.Weletz 1 ∼kcisθ 1 ,z 2 ∼kcisθ 2 ,w 1 ∼kcisφ 1 ,w 2 ∼kcisφ 2 .ThenZ 1 W 2
andW 1 Z 2 are parallel if and only if

kcisφ 2 −kcisθ 1

kcisθ 2 −kcisφ 1

=t