216 Vector and complex-number methods Ch. 11
Next suppose thatz 1 =z 2 +(p 1 +q 1 ı)(z 3 −z 2 ),z′ 1 =z′ 2 +(p 1 −q 1 ı)(z′ 3 −z′ 2 ).By
an analogous argument the triangles are still similar, and now the triples(Z 1 ,Z 2 ,Z 3 )
and(Z 1 ′,Z 2 ′,Z 3 ′)are oppositely oriented.
Conversely, suppose that[Z 1 ,Z 2 ,Z 3 ]and[Z 1 ′,Z 2 ′,Z′ 3 ]are similar triangles in the
correspondence(Z 1 ,Z 2 ,Z 3 )→(Z 1 ′,Z 2 ′,Z 3 ′).LetW 1 be the foot of the perpendicular
fromZ 1 toZ 2 Z 3 , and from parametric equations ofZ 2 Z 3 choosep 1 ∈Rso thatw 1 =
z 2 +p 1 (z 3 −z 2 ).Then
|Z 2 ,W 1 |
|Z 2 ,Z 3 |
=|p 1 |,andp 1 is positive or negative according or not asW 1 is on the same or opposite side
ofZ 2 asZ 3 is on the lineZ 2 Z 3 , that is according as the wedge-angle∠Z 3 Z 2 Z 1 is acute
or obtuse. AsW 1 Z 1 ⊥Z 2 Z 3 we can findq 1 ∈Rso thatz 1 −w 1 =q 1 ı(z 3 −z 2 ).Then
|Z 1 ,W 1 |
|Z 2 ,Z 3 |=|q 1 |,andq 1 is positive or negative according as(Z 1 ,Z 2 ,Z 3 )is positively or negatively
oriented.
As the lengths of the sides of the two triangles are proportional, we have
|Z 2 ′,Z 3 ′|=k|Z 2 ,Z 3 |,|Z 3 ,Z 1 |=k|Z 3 ′,Z 1 ′|,|Z 1 ′,Z′ 2 |=k|Z 1 ,Z 2 |,for somek>0. LetW 1 ′be the foot of the perpendicular fromZ 4 to the lineZ 5 Z 6 .Then
the triangles[Z 1 ,Z 2 ,W 1 ]and[Z 1 ′,Z′ 2 ,W 1 ′]are similar, so we have that
|Z′ 2 ,W 1 ′|
|Z 2 ,W 1 |=
|Z 1 ′,Z′ 2 |
|Z 1 ,Z 2 |
=k.It follows that
|Z 2 ′,W 1 ′|=k|Z 2 ,W 1 |=k|p 1 ||Z 2 ,Z 3 |=|p 1 ||Z′ 2 ,Z 3 ′|.ButW 1 ′is on the same side of the pointZ 2 ′on the lineZ 2 ′Z 3 ′asZ 3 ′is if the wedge-angle
∠Z′ 3 Z 2 ′Z 1 ′is acute, and on the opposite side if this angle is obtuse. Hence we have that
w′ 1 −z′ 2 =p 1 (z′ 3 −z 2 ).
AsZ′ 1 W 1 ′⊥Z 2 ′Z 3 ′,wehavez′ 1 −w′ 1 =jı(z′ 3 −z′ 2 )for somej∈R,andthen|Z 1 ′,W 1 ′|=
|j||Z′ 1 ,Z 3 ′|.But
|Z 1 ′,W 1 ′|
|Z 1 ,W 1 |
=
|Z 1 ′,Z 2 ′|
|Z 1 ,Z 2 |
=
|Z′ 2 ,Z′ 3 |
|Z 2 ,Z 3 |
,
so
|Z 1 ′,W 1 ′|
|Z 2 ′ 5 ,Z 3 ′|
=
|Z 1 ,W 1 |
|Z 2 ,Z 3 |
=|q 1 |.Hencej=±q 1 and we are to take the plus if(Z 1 ,Z 2 ,Z 3 )and(Z′ 1 ,Z 2 ′,Z 3 ′)have the
same orientation, the minus if the opposite orientation.
Thus our mobile coordinates(p 1 ,q 1 )are intimately connected with similarity of
triangles.