Mathematical_Recreations-Kraitchik-2e

Arithmetico-Geometrical Questions 107
In addition there are four Heronian parallelograms (parallelograms
with rational sides, diagonals, and areas):
ACA'C', ADA'D', BCB'C', BDB'D'.
FIGURE 16. Diagonals and Edges of Three Faces of a Cuboid.
We may obtain the same result by starting with two arithmetical
angles, A, B, given by
P2 _ q2
sin A = "--.,----=-=
p2 + q2
. r 2 - 8 2
slnB= --­
r 2 + 8 2
or
or
2pq
p2 + q2'
2 r8
r 2 + 8 2 •
When any pair of values of sin A and sin B has been selected,
the larger acute angle may be taken as interior or exterior to
the triangle, so eight different triangles may be formed.
Again, we may start with the circumscribed circle of radius
R, using the relations
abc
sin A = sin B = sin C = 2R.