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

36] THE SERIES OF ASSOCIATED TRIANGLES 65

The relation between the elements of these triangles can be
put in the form of the following rule:
Let a, b, c, (X = (1), / = II (m)) be the sides, hypothenuse, and
the angles opposite the sides of a right-angled triangle. Write the
letters a', 1, c, m, b' in cyclic order on the sides of a pentagon. The
six triangles which form the closed series of associated triangles
are obtained, if we write the letters a,', lr, Cr, mr, br' in the same or
reverse order on the sides, starting with any one side, and take the
elements with the suffices equal to those on the same sides without
the suffices.
C1
C r
r I
C,
FIG. 44.
E.g. from Fig. 44,
a,' =1, i.e. a,= ';
b/ = a', i.e. b, = a;
mr =b', i.e.,=-r- _;
C, = Tm;
l.=c, i.e. X.=y;
giving the triangle (5) above.
These results have an important bearing on certain problems
of construction. For example, the problem of constructing a
right-angled triangle when the hypothenuse c and a side a are
given, with the usual construction involves the assumption as
N.-E.G. E