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

84

NON-EUCLIDEAN GEOMETRY

[CH. III.

is a Limiting-Curve, or Circle of Infinite Radius, with its centre
at the vertex of the pencil.
(c) The locus of corresponding points upon a pencil of lines,
whose vertex is an improper point-an ideal point-is an Equidistant-Curve, whose base-line is the representative line of the
ideal point.
According as the perpendiculars to the sides of a triangle ABC at
their middle points meet in an ordinary point, a point at infinity, or an
ideal point, the points ABC determine an ordinary circle, a limiting-curve,
or an equidistant-curve. (Cf. ~ 39.)
THE MEASUREMENT OF AREA.
~50. Equivalent Polygons.
Two polygons are said to be equivalent when they can be broken
up into a finite number of triangles congruent in pairs.
3 6/
\o 3
P' P3
P2.
FIG. 59.
With this definition of equivalence, we shall now prove the
following theorem:
If two polygons P, and P2 are each equivalent to a third polygon
P3, then P1 and P2 are equivalent to each other.