11 Vector and complex-number methods
11.1 Equipollence..............................
11.1.1 .....................................
Definition. An ordered pair(Z 1 ,Z 2 )of points inΠissaidtobeequipollentto the pair
(Z 3 ,Z 4 ), written symbolically(Z 1 ,Z 2 )↑(Z 3 ,Z 4 ),ifmp(Z 1 ,Z 4 )=mp(Z 2 ,Z 3 ). Thus↑
is a binary relation inΠ×Π.
Equipollence has the properties:-(i)If Z 1 ≡(x 1 ,y 1 ),Z 2 ≡(x 2 ,y 2 ),Z 3 ≡(x 3 ,y 4 ),Z 4 ≡(x 4 ,y 4 ),then
(Z 1 ,Z 2 )↑(Z 3 ,Z 4 )if and only if x 1 +x 4 =x 2 +x 3 ,y 1 +y 4 =y 2 +y 3 ,or equiv-
alently x 2 −x 1 =x 4 −x 3 ,y 2 −y 1 =y 4 −y 3.(ii) Given any points Z 1 ,Z 2 ,Z 3 ∈Π, there is a unique point Z 4 such that(Z 1 ,Z 2 )↑
(Z 3 ,Z 4 ).(iii) For all Z 1 ,Z 2 ∈Π,(Z 1 ,Z 2 )↑(Z 1 ,Z 2 ).(iv) If(Z 1 ,Z 2 )↑(Z 3 ,Z 4 )then(Z 3 ,Z 4 )↑(Z 1 ,Z 2 ).(v)If(Z 1 ,Z 2 )↑(Z 3 ,Z 4 )and(Z 3 ,Z 4 )↑(Z 5 ,Z 6 ),then(Z 1 ,Z 2 )↑(Z 5 ,Z 6 ).(vi)If(Z 1 ,Z 2 )↑(Z 3 ,Z 4 )then(Z 1 ,Z 3 )↑(Z 2 ,Z 4 ).(vii) If(Z 1 ,Z 2 )↑(Z 3 ,Z 4 ),then|Z 1 ,Z 2 |=|Z 3 ,Z 4 |.(viii) For all Z 1 ∈Π,(Z 1 ,Z 1 )↑(Z 3 ,Z 4 )if and only if Z 3 =Z 4.
(ix)If Z 1 =Z 2 and Z 3 ∈l=Z 1 Z 2 ,then(Z 1 ,Z 2 )↑(Z 3 ,Z 4 )if and only Z 4 ∈l,
|Z 1 ,Z 2 |=|Z 3 ,Z 4 |and if≤lis the natural order for which Z 1 ≤lZ 2 ,thenZ 3 ≤l
Z 4.Geometry with Trigonometry
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