The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

60

NON-EUCLIDEAN GEOMETRY

[CH. I1I.

Now draw through A the parallel to the line BC (Fig. 39).
Also draw the line perpendicular to c, which is parallel to
BC in the same sense. This line will cut the hypothenuse, or
the hypothenuse produced, according as m is less than or
greater than c.
If mn < c, we have
X + - = n (c- n?).........................(2)
If m > c, then we would have
7r - X I (n - c).
A
1~^\ ~* A
b
m <B _a C
FIG. 39. FIG. 40.
With the usual notation (cf. ~ 27) this reduces to
X + =n(c -m).
In the same way we have
+ = (c -........................(2')
Finally, produce CB through B, and draw the perpendicular
to CB produced which is also parallel to AB (Fig. 40). Also
produce AC through C, and draw the perpendicular to AC
which is parallel to AB.
From Fig. 40, if we suppose a line drawn through C parallel
to AB, it is clear that
n(I - b) + n (m + a)=,..................(3)

and similarly

ad(m -a)si l+al) = 2............ (3')