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

51] THE AREA OF A TRIANGLE

87

Construct the right-angled triangle with a side equal to CC'
and ~b1 for the hypothenuse.*
Cut off C'E2 equal to the other side of this triangle.
Join CE2, and produce it to A2 making CE =E2A2.
Join A2B.
Then the triangle A2BC has a side equal to b1, and the same
defect as the two given triangles.
Also the triangles ABC and A2BC are equivalent; and the
triangles A2BC and A1B1C1, by (1).
Therefore the triangles ABC and A1B1C1 are equivalent (~ 50).
3. Any two triangles, with the same defect, are equivalent.
For a side of one must be greater than, equal to, or less
than, a side of the other.
When it is a case of equality, the triangles are equivalent
by (1).
In the other two cases, the same result follows from (2).
4. The converse of this theorem also holds:
Any two equivalent triangles have the same defect.
From the definition of equivalence, the two triangles can be
broken up into a finite number of triangles congruent in pairs.
But if a triangle is broken up by transversals t into a set of
sub-triangles, it is easy to show that the defect of the triangle
is equal to the sum of the defects of the triangles in this partition. Further, following Hilbert,+ it can be shown that any
given partition of a triangle into triangles can be obtained by
successive division by transversals. It follows that the sum of
the defects of the triangles is equal to the defect of the original
triangle.
Now the two equivalent triangles can be broken up into a
finite number of triangles congruent in pairs. And the defects
of congruent triangles are equal.
* The construction of the right-angled triangle from a side and the
hypothenuse does not involve the Principle of Continuity. The results
of ~ 36 show that this problem can be reduced to that of constructing a
right-angled triangle out of a side and the adjacent angle.
t A triangle is said to be broken up by transversals, when the partition into triangles is obtained by lines from the angular points to the
opposite sides, either in the original triangle or in the additional
triangles which have been obtained from the first by division by
transversals.
+ Cf. Hilbert, loc. cit. ~ 20, or Halsted, Rational Geometry, p. 87.