174 Complex coordinates; sensed angles; angles between lines Ch. 10
so that
y 6
x 6
=
2 x 4 (k−y 4 )
(k−y 4 )^2 −x^24Then
y 6
x 6
y 4
x 4= 2
y 4 (k−y 4 )
k^2 − 2 ky 4 +y^24 −x^24= 2
y 4 (k−y 4 )
2 y^44 − 2 ky 4=− 1.
10.10.7 Associativity of addition of duo-angles ...............
With the notation of 10.10.5, suppose thatαd,βdandγdare duo-angles inDA(F),
with pairs side- lines(OQ,OZ 4 ),(OQ,OZ 5 ),(OQ,OZ 6 ), respectively. We wish to
consider the sums(αd+βd)+γdandαd+(βd+γd). We suppose thatαd+βdhas
side-lines(OQ,OZ 7 )and that(αd+βd)+γdhas side-lines(OQ,OZ 9 ). Similarly we
suppose thatβd+γdhas side-lines(OQ,OZ 8 )andαd+(betad+γd)has side-lines
(OQ,OZ 10 ). Then by (10.10.2) applied several times we have that
y 7 =x 5 y 4 +x 4 y 5
k,x 7 =x 4 x 5 −y 4 y 5
k,
y 9 =x 6 y 7 +x 7 y 6
k=
x 6 x^5 y^4 +kx^4 y^5 +x^4 x^5 −ky^4 y^5 y 6
k
=x 6 (x 5 y 4 +x 4 y 5 )+(x 4 x 5 −y 4 y 5 )y 6
k^2,
x 9 =x 7 x 6 −y 7 y 6
k=
x 4 x 5 −y 4 y 5
k x^6 −x 5 y 4 +x 4 y 5
k y^6
k
=(x 4 x 5 −y 4 y 5 )x 6 −(x 5 y 4 +x 4 y 5 )y 6
k^2.
Similarly
y 8 =x 6 y 5 +x 5 y 6
k,x 8 =x 5 x 6 −y 5 y 6
k,
y 10 =x 8 y 4 +x 4 y 8
k=
x 5 x 6 −y 5 y 6
k y^4 +x^4x 6 y 5 +x 5 y 6
k
k
=(x 5 x 6 −y 5 y 6 )y 4 +x 4 (x 6 y 5 +x 5 y 6 )
k^2,
x 10 =x 4 x 8 −y 4 y 8
k =x 4 x^5 x^6 −ky^5 y^6 −y 4 x^6 y^5 +kx^5 y^6
k
=x 4 (x 5 x 6 −y 5 y 6 )−y 4 (x 6 y 5 +x 5 y 6 )
k^2.
From these we can see thatZ 9 =Z 10 and so we have that(αd+βd)+γd=αd+(βd+γd).Thus addition of duo-angles is associative onDA(F).