Let the sides of the triangle be a, b, and c. Using Heron's formula for area, we get
(b+c-a)(c+a-b)(a+b-c) = 16(a+b+c) (1)
Let
l = b+c-a, m = c+a-b, n = a+b-c (2)
so that
a = (m+n)/2, b = (n+l)/2, c = (l+m)/2 (3)
From (2) and the triangle inequalities, we know that l, m, and n are positive integers. From (1) and (3) we know that they all must be even. Let
l = 2p, m = 2q, n = 2r (4)
so that
a = q+r, b = r+p, c = p+q (5)
Combining equations (1) and (5) gives
pqr = 4(p+q+r) (6)
WOLOG let
p >= q >= r >= 1 (7)
Solving (6) for p gives
4(q+r)
p = -------- (8)
qr-4
(7) and (8) imply
5 8q
max(1,---) <= r <= min(q,------) , for q>0
q q2-4
Therefore, we only have to test seven q,r pairs
for integer p. The following table gives the "five"
triangles
p q r a b c Area = Perimeter Type
----------------------------------------------------
10 3 2 5 12 13 30 Right
6 4 2 6 8 10 24 Right
24 5 1 6 25 29 60 Obtuse
14 6 1 7 15 20 42 Obtuse
9 8 1 9 10 17 36 Obtuse
I have found one place on the WEB where these triangles
are called perfect triangles.
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