Geometry with Trigonometry

178 Complex coordinates; sensed angles; angles between lines Ch. 10

Figure 10.16. Very symmetrical cases.

Proof. For from the first two we deduce thatZ 1 W 3 ‖W 1 Z 3 and on combining this
with the third relation we obtain the conclusion.

NOTE. Clearly this last result can be extended to any number of pairs of points on
a circle.

10.11.2

Starting more generally than in the last subsection, for pairs of distinct points let
(Z 1 ,W 1 )∼(Z 2 ,W 2 )if and only ifZ 1 W 2 ‖W 1 Z 2. Then clearly the relation∼is reflexive
and symmetric. We ask when it is also transitive and thus an equivalence relation.
Now ifZ 1 ≡(x 1 ,y 1 ),Z 2 ≡(x 2 ,y 2 ),W 1 ≡(u 1 ,v 1 ),W 2 ≡(u 2 ,v 2 ),wehave
(Z 1 ,W 1 )∼(Z 2 ,W 2 )if and only if

(v 2 −y 1 )(x 2 −u 1 )=(u 2 −x 1 )(y 2 −v 1 ). (10.11.1)

Similarly we have(Z 2 ,W 2 )∼(Z,W)if and only if

(v−y 2 )(x−u 2 )=(u−x 2 )(y−v 2 ). (10.11.2)

We wish (10.11.1) and (10.11.2) to imply that

(v−y 1 )(x−u 1 )=(u−x 1 )(y−v 1 ). (10.11.3)

From (10.11.1) we have that

v 2 x 2 −u 2 y 2 =u 1 v 2 −u 2 v 1 +x 2 y 1 −x 1 y 2 +x 1 v 1 −y 1 u 1 ,

and from (10.11.2)

v 2 x 2 −u 2 y 2 =uv 2 −u 2 v+x 2 y−xy 2 +vx−uy,

so together these give

vx−uy=u 1 v 2 −u 2 v 1 +x 2 y 1 −x 1 y 2 +x 1 v 1 −y 1 u 1 −uv 2 +u 2 v−x 2 y+xy 2.

We need for (10.11.3) that

vx−uy=vu 1 −uv 1 +y 1 x−x 1 y−y 1 u 1 +x 1 v 1