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

35, 36] TRIANGLE AND QUADRILATERAL

63

But from (I.) and (II.) we have
\1 + i = II( - m,)= nI (c - n),
- A + / 3 U= I(cl + m) = II(c + ).
Therefore X = X and /3i=/.
From (III') and (III.), we now obtain
H (v a - a)= - ( + b),
H n(n, - a,) = n(n - a) = - H(11 + bl) - H n( + b)
Thus m - aC = - ta,
and a1 = a.
Therefore we have obtained the important result:
If a,, b, (X, (X,) are the five elements of a right-angled triangle,
then there exists a quadrilateral with three right angles and one
acute angle, in which the sides are c, m', a, and 1, taken in order,
and the acute angle 3S lies between c and 1.*
The converse of this theorem also holds.
~ 36. The Closed Series of Associated Right-Angled
Triangles.
We have seen that to the right-angled triangle a, b, c, (X,,u)
there corresponds a quadrilateral with three right angles and
* This result was given by Lobatschewsky in his earliest work, On the
Principles of Geometry (cf. ~~ 11, 16, Engel's translation, pp. 15 and 25),
but his demonstration requires the theorems of the Non-Euclidean Solid
Geometry. The proof in the text is due to Liebmann (Mlith. Ann.
vol. lxi. p. 185 (1905), and Nichte7nklidische Geomnetrie, 2nd ed. ~ 10),
who first established the correspondence between the right-angled triangle and the quadrilateral with three right angles and an acute angle
by the aid of Plane Geometry alone.
This is an important development, as the Parallel Constructions
depend upon this correspondence, and the Non-Euclidean Plane Geometry
and Trigonometry is now self-contained.
Further, as we shall see below (~ 45), the existence of a segment
corresponding to any given angle of parallelism can be established
without the use of the Principle of Continuity, on which Lobatschewsky's
demonstration depends. Therefore, though the existence of p, when
1(p) is given, is assumed in the above demonstration, the correspondence between the triangle and quadrilateral is independent of that
principle.