Mathematical_Recreations-Kraitchik-2e

104 Mathematical Recreations
(2ef, e2 - p, e2 + P) and (2gh, g2 - h2, g2 + h2),
where e2 + P = (ae + bd)2 + (ad - bc)2
and g2 + h2 = Cae - bd)2 + (ad + bC)2.
If the smaller angle of (x, y, z) is greater than the smaller
angle of (x', y', z'), then it can easily be shown that the
smaller angle of the first of these new triangles is the difference
of the smaller angles of the original triangles, and the smaller
angle of the second is their sum.
If z = z', then ad - be = 0 and only the second triangle
exists. In this case the smaller angle of the new triangle is
twice the smaller angle of the original.
13. HERONIAN TRIANGLES. A Heronian, or arithmetical,
triangle is one whose sides and area are all rational. Hence
the altitudes are also rational. The triangle is accordingly
called by some mathematicians a rational triangle. If the
vertices are A, B, C, the lengths of the opposite sides are a,
h, c, the semiperimeter Ha + b + e) is 8, the area is K, and
the radius of the inscribed circle is r, then we know from geometry
that
K
B r C r
A r
r= - tan -=--
8 ' 2 8 -a'
tan 2" = 8 -
b'
tan -=--
2 8 - e'
so these quantities are all rational and the angles are
arithmetical. One can also show quite easily that the radii
of the circumscribed and escribed circles are rational.
Since the angles of a Heronian triangle are arithmetical,
a perpendicular from a vertex to an opposite side forms two
rational right triangles whose sum or difference is the given
triangle. By using this fact we can form any desired Heronian
triangle which is not a right triangle by combining two
rational right triangles of different shapes. Let (x, y, z) =
(2ab, a 2 - b 2 , a 2 + b 2 ) and Cu, v, w) = (2cd, c2 - d 2 , c2 + d 2 )
be two such primitive triangles. If we form triangles similar