234 Vector and complex-number methods Ch. 11
11.8 Isogonal conjugates ..........................
11.8.1 Isogonal conjugates
Definition.Given non-collinear points Z 1 ,Z 2 ,Z 3 , we say that half-lines[Z 1 ,Z 4
[Z 1 ,Z 5 areisogonal conjugateswith respect to the angle-support|Z 2 Z 1 Z 3 if the
sensed anglesFZ 2 Z 1 Z 4 ,FZ 5 Z 1 Z 3 , have equal magnitudes.
Z 1
Z 2
Z 3
Z 4
Z 5
Figure 11.21. Isogonal conjugates.
To see how this operates, we first suppose thatZ 4 andZ 5 are both on the lineZ 2 Z 3
and that
Z 4 =
1
1 +λ 1
Z 2 +
λ 1
1 +λ 1
Z 3 ,Z 5 =
1
1 +λ 2
Z 2 +
λ 2
1 +λ 2
Z 3 ,
for real numbersλ 1 andλ 2. We recall that then
Z 2 Z 4
Z 4 Z 3
=λ 1 ,
Z 2 Z 5
Z 5 Z 3
=λ 2.
Then
z 4 −z 1
z 2 −z 1
/
z 3 −z 1
z 5 −z 1
=
1
1 +λ 1 z^2 +
λ 1
1 +λ 1 z^3 −z^1
z 2 −z 1
1
1 +λ 2 z^2 +
λ 2
1 +λ 2 z^3 −z^1
z 3 −z 1
is positive and so on multiplying across by( 1 +λ 1 )( 1 +λ 2 )
z 2 −z 1 +λ 1 (z 3 −z 1 )
z 2 −z 1
z 2 −z 1 +λ 2 (z 3 −z 1 )
z 3 −z 1
=
z 2 −z 1
z 3 −z 1
+λ 1 λ 2
z 3 −z 1
z 2 −z 1
+λ 1 +λ 2
is real. On subtractingλ 1 +λ 2 it follows that
z 2 −z 1
z 3 −z 1
+λ 1 λ 2
1
(z 2 −z 1 )/(z 3 −z 1 )
is real. We write
z 2 −z 1
z 3 −z 1
=u+vı, (u,v∈R),