Geometry with Trigonometry

Sec. 10.10 Angles between lines 171

10.10.5 Addition of duo-angles in standard position .....

To deal with addition of duo-
angles in standard position, let
Q≡(k, 0 ),R≡( 0 ,k)for some
k>0, andαd have side-lines
OQandOZ 4 ,βdhave side-lines
OQandOZ 5 ,where|O,Z 4 |=
|O,Z 5 |=kand both have their
indicators inD 1. Without loss of
generality, we may suppose that
eithery 4 >0ory 4 = 0 ,x 4 >
0, and similarly with respect to
(x 5 ,y 5 ).

Z 4

Z 5

Z 6

O

J Q

I

R

S

T

αd

βd

D 1

D 2

D 1

D 2

Figure 10.14. Addition of duo-angles.

Then the line throughQwhich is parallel toZ 4 Z 5 will meet the circleC(O;k)in a
second point, which we denote byZ 6 ≡(x 6 ,y 6 ). The line throughQparallel toZ 4 Z 5
has parametric equations

x=k+t(x 5 −x 4 ),y=t(y 5 −y 4 ),

and so meets the circle again whent=0 satisfies

[k+t(x 5 −x 4 )]^2 +[t(y 5 −y 4 )]^2 =k^2.

This yields

t=−

2 k(x 5 −x 4 )

(x 5 −x 4 )^2 +(y 5 −y 4 )^2

,

and so we find for(x 6 ,y 6 )that

x 6 =k

(y 5 −y 4 )^2 −(x 5 −x 4 )^2

(x 5 −x 4 )^2 +(y 5 −y 4 )^2

,y 6 =− 2 k

(x 5 −x 4 )(y 5 −y 4 )

(x 5 −x 4 )^2 +(y 5 −y 4 )^2

. (10.10.1)

We define thesumαd+βd=γd,whereγdhas side-linesOQandOZ 6 and has its
indicator inD 1 .WhenZ 4 =Z 5 we takeQZ 6 as the line throughQwhich is parallel to
the tangent to the circle atZ 4. This is analogous to the modified sum of angles.
It can be checked that
y 6
k

−

x 5 y 4 +x 4 y 5

k^2

= 0 ,

x 6

k

−

x 4 x 5 −y 4 y 5

k^2

= 0. (10.10.2)

To see this we first note that(y 5 −y 4 )^2 +(x 5 −x 4 )^2 = 2 [k^2 −(x 4 x 5 +y 4 y 5 )]. Then the
numerator in
(x 5 −x 4 )(y 5 −y 4 )
x 4 x 5 +y 4 y 5 −k^2

−

x 5 y 4 +x 4 y 5

k^2