I first discovered this problem in a posting to the NMBRTHRY listserver from Jim Buddenhagen.

Assume that the sides of a triangle are a,b,c with corresponding opposite angles A,B,C.

Then the cosine rule gives

c2  =  a2 + b2 - 2 a b cos C

and, if  area = g²,

a b sin C  =  2 g2

Assume that tan(C/2) = n / d with n,d integers with no common factors. Then

sin C =  2 n d n2 + d2           ,         cos C =  n2 - d2 n2 + d2

If we define a / g = j,  c / g = j,  p = d n j,  q = d² n² k j,   then

q2  =  p4  +  2 d n ( n2 - d2 ) p2  +  d2 n2 ( d2 + n2 )2

This quartic has an obvious rational point, so is birationally equivalent to an elliptic curve. Using standard methods, and some algebraic manipulation, we find the family of curves

s2  =  t  ( t - d n3 )  ( t + d3 n )

with p = s / t and  ±q = d n ( d² - n² ) + 2 t  -  s² / t².

As an example, let n = 2,  d = 1. Then s = 3, t = -1 lies on the curve s² = t ( t - 8 ) ( t + 2 ),

so p = 3,   j = 3/2,  q = 17,  k = 17/6,   giving a = 9,  c = 17,  b = 10 and area = 36.

There are clearly 4 points of order 2,  t = 0, t = d n³, t = -d³ n, and the point at infinity. I can show that there are no points of order 4, but am unable to prove that the torsion subgroup is Z2×Z2 and not Z2 ×Z6, though experimental evidence suggests this is true. Anybody out there help?

The 4 points of order 2 do not lead to a triangle, so, if these are the only torsion points, a triangle only exists if the rank of the curve is > 0.

The Birch and Swinnerton-Dyer conjecture was then used to estimate the rank for the cases
(a) 1 <= n <= 99  ,  d = 1 ,
(b) n = 1  ,  1 <= d <= 99,
with the results in the following tables.
 

                                         Rank of curve for 1 <= n <= 99  ,  d = 1
 

                                         Rank of curve for n = 1 , 1 <= d <= 99
 
 

For most of the curves of rank > 0, a rational point has been found, and hence a triangle satisfying the criteria of the problem. Readers interested in these solutions can find the results in the following links (zero values indicate no known solution):

(a) results for 1 <= n <= 99 , d = 1 ,

(b) results for n = 1 , 1 <= d <= 99.
 

It should be pointed out that there are 9 unsolved curves in the first group, but only 2 in the second. Some of the predicted heights of the rational points in the first group are huge. The list of unsolved curves is in the following table.