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

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NON-EUCLIDEAN GEOMETRY

[CH. IV.

Then MW2 is perpendicular to the chord BB', and is symmetrical to the two parallels AB and A'B' (cf. ~ 26 (4) and ~ 47).
Therefore it passes through the middle point N of BB'.
Then it follows from the quadrilateral ABB'A' that AB = A'B'.
2. If A, B and A', B' are the points in which two Concentric
Limiting-Curves cut two of their axes, and P, Q are the middle
points of the arcs AA' and BB', then PQ is a line of the pencil.
A
oB
Al
FIG. 63.
Since equal arcs subtend equal chords (cf. ~ 48), the chords
AP and A'P are equal, and the chords BQ and B'Q are equal.
It follows that PQ is the line of symmetry for the two axes
AB and A'B', and is parallel to both (cf. ~ 47).
COROLLARY. If the points P1, P2, P3, P4,..., divide the arc AA'
into n equal arcs, and the axes through these points are met by the
Limiting-Curve BB' in Q1,, Q2, Q, 4,..., the points Q1, Q2,
Q3, Q4,... divide the arc BB' into n equal arcs.
3. If A, A', A", are three points on a Limiting-Curve, and
B, B', B", are the three points in which a Concentric LimitingCurve is cut by the axes through A, A', and A", then
arc AA': arc AA"=arc BB': arc BB".
First, let the arcs AA' and AA" be commensurable, and let
the one be m times the arc AP and the other n times the arc
AP.
Through P draw the line of the pencil. Let it cut the second
Limiting-Curve in Q.