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

62

NON-EUJCLID)EAN GEOMETRY c

[cii. iri.

Finally, from QP cut off the segment in, and from PS produced the segment bl, and raise the perpendiculars at the ends
of these lines (Fig. 43).
bi
S R
13I
p
FIG. 43.
It follows that
7in 11, + b,) + I, - aI,)I............... )
and correspondingly
IT(c1 + b1) + fl(a,' - fin1') =. (III'.)
III. We are now able to establish the correspondence between the two figures.
A right-angled triangle is fully determined when we know
c and; a quadrilateral of this nature, when we know c,
and in1'.
7r~
Let C1=c and IT(m,')= laL
so that ml = M.
Then it follows from (1') and (2) that
X ~/3p=f(C - 9n),
-X.4-=8 c +fnm),
and. therefore 2X H (c - vn) - H(c + in),

2/3 = fl (c- in) + ~f(c +vn).