Geometry with Trigonometry

208 Vector and complex-number methods Ch. 11

Given non-collinear pointsZ 1 ,Z 2 ,Z 3 , by 11.4.1 we can express

−−→

OZ 1 =

−−→

OZ 2 +p 1 (

−−→

OZ 3 −

−−→

OZ 2 )+q 1 (

−−→

OZ 3 −

−−→

OZ 2 )⊥,

for unique real numbersp 1 andq 1 , not both equal to 0. This is the geometrical back-
ground but the manipulations are simpler if we use complex coordinates instead.
In (11.6.1) for anyZ=Owe takeZ∼zwherez=x+ıyandW∼wwhere
w=u+iv. Then we note thatız=−y+ıxand soW=Z⊥if and only ifw=ız.Then
forZ 1 ∼z 1 ,Z 2 ∼z 2 ,Z 3 ∼z 3 we write

z 1 −z 2 =p 1 (z 3 −z 2 )+q 1 ı(z 3 −z 2 )=(p 1 +q 1 ı)(z 3 −z 2 ), (11.6.1)

that is

(x 1 −x 2 ,y 1 −y 2 )=(p 1 (x 3 −x 2 )−q 1 (y 3 −y 2 ),p 1 (y 3 −y 2 )+q 1 (x 3 −x 2 )).

We coin the namemobile coordinatesof the pointZ 1 with respect to(Z 2 ,Z 3 )and
F,forthepair(p 1 ,q 1 ).

It follows immediately that|Z 1 ,Z 2 |=

√

p^21 +q^21 |Z 2 ,Z 3 |, and as from (11.6.1)z 1 −

z 3 =(p 1 − 1 +q 1 ı)(z 3 −z 2 )we also have|Z 3 ,Z 1 |=

√

(p 1 − 1 )^2 +q^21 |Z 2 ,Z 3 |.From

z 3 −z 1

z 2 −z 1

=

p 1 − 1 +q 1 ı

p 1 +q 1 ı

,

z 1 −z 2

z 3 −z 2

=p 1 +q 1 ı,

z 2 −z 3

z 1 −z 3

=

1

1 −p 1 +q 1 ı

,

withα=FZ 2 Z 1 Z 3 ,β=FZ 3 Z 2 Z 1 ,γ=FZ 1 Z 3 Z 2 ,wehavethat

cisα=√(p^1 −^1 +q^1 ı)(p^1 −q^1 ı)

p^21 +q^21

√

(p 1 − 1 )^2 +q^21

,cisβ=√p^1 +q^1 ı

p^21 +q^21

,cisγ=√^1 −p^1 +q^1 ı

( 1 −p 1 )^2 +q^21

.

We also have that

p 1 +ıq 1 =

√

p^21 +q^21 cisβ=

c

a

cisβ,

1 −p 1 −ıq 1 =

√

(p 1 − 1 )^2 +q^21 cisγ=

b

a

cisγ,

p 1 (p 1 − 1 )+q^21 +ıq 1 =

√

p^21 +q^21

√

(p 1 − 1 )^2 +q^21 cisα

=

bc

a^2

cisα. (11.6.2)

Thus we have in terms ofp 1 andq 1 , the ratios of the lengths of the sides and cosines
and sines of the angles. Moreover, it is easily calculated that

δF(Z 1 ,Z 2 ,Z 3 )=

q 1

2

|Z 2 ,Z 3 |^2 ,

so the orientation of this triple is determined by the sign ofq 1.