Geometry with Trigonometry

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),