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

41, 42, 43] THE PARALLEL CONSTRUCTIONS

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~ 43. Second Proof of Bolyai's Parallel Construction.
The following proof of the validity of Bolyai's construction is due to Liebmann: * it will be seen that it depends
(1) on Theorem (2) of ~ 4, regarding the locus of the middle
points of the segments AA', BB', etc., joining a set of
points, A, B, C,..., A', B', C',..., on two straight lines,
such that AB= A'B', BC =B'C', etc.; and (2) on the concurrence of the perpendicular bisectors of the sides of a
triangle (cf. ~ 39).
Let A be the given point, and AF the perpendicular from A
to the given line.
It is required to draw from A the parallel to the ray FQ2.
Let us suppose the parallel A2 drawn.
From A2 and F2 cut off equal segments AS and FD, and
join SD.
Let M and M' be the middle points of AF and SD.
From ~4 we know that the line MM' is parallel to A2
and Fi2.
<_ _ A B __ __
FIG 49.f
F,
FIG. 49.
Draw the line 2"AM' through A perpendicular to AF, and
produce M'M through the point M.
Then it is clear that the ray M'M is parallel to the line A'".
Draw from F the parallel FQ' to AM2', and let it intersect AQ
in G.
From F2' cut off FS' equal to AS. Join SS' and S'D.
The line GM bisects SS' at right angles, and is perpendicular
to the line 22'.
Also the perpendicular bisector of DS' bisects the angle DFS',
and is perpendicular to 22'.
* Ber. d. k. sachs. Ges. d. Wiss. Math. Phys. Klasse, vol. lxii. p. 35
(1910); also Nichteuklidische Geometrie (2nd ed.), p. 35.