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.