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

68

NON-EUCLIDEAN GEOMETRY

[CH. III.

(6) Two ideal points [rP, Pr], when the lines c and c' do not
intersect and are not parallel. The line rFr,, is the common
perpendicular to the two not-intersecting lines c and c'. The
construction of this line was given in ~ 32.
The pairs of points which do not determine a line are as
follows:
(i) An ideal point and a point at infinity lying on the representative line of the ideal point.
(ii) Two ideal points, whose representative lines are parallel
or meet in an ordinary point.*
~ 39. With this notation the theorems as to the concurrence
of the lines bisecting the sides of a triangle at right angles, the
lines bisecting the angles of a triangle, the perpendiculars from
the angular points to the opposite sides, which hold in the
Euclidean Geometry, will be found also to be true in this NonEuclidean Geometry. Lines will be said to intersect in the
sense of ~~37, 38. Also, in speaking of triangles, it is not
always necessary that they should have ordinary points for
their angular points. The figure of ~ 26 is a triangle with one
angular point at an improper point —a point at infinity. It
will be seen that a number of the theorems of that section
are analogous to familiar theorems for ordinary triangles.
With regard'to the concurrence of lines in the triangle we
shall only take one case-the perpendiculars through the
middle points of the sides.
The perpendiculars to the sides of a triangle at their middle
points are concurrent.
Let ABC be the triangle and D, E, F the middle points of the
sides opposite A, B and C.
Case (i) If the perpendiculars at the middle points of two
of the sides intersect in an ordinary point, the third perpendicular must also pass through this point. The proof depends
on the congruence theorems as in the Euclidean case.
*In the foundation of Projective Geometry independent of the
Parallel Postulate, this difficulty is overcome by the introduction of
new entities, called improper lines, and ideal lines, to distinguish them
from the ordinary or proper lines. Cf. Bonola, loc. cit. English translation, App. IV.