Find all the non-trivial integer solutions to the equation
$$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=4.$$



Solution to a Diophantine equation
FWIW, using Michael Stoll' answer to [Estimating the size of solutions of a diophantine equation](https://mathoverflow.net/questions/227713/estimating-the-size-of-solutions-of-a-diophantine-equation), we use Magma to do:



```
F := EllipticCurve(x^3 + 109* x^2 + 224 * x);
IntegralPoints(F);
[ (-100 : -260 : 1), (-56 : 392 : 1), (-9 : 78 : 1), (-4 : -28 : 1), (0 : 0 : 1), (4 : -52 : 1), (56 : 
-728 : 1) ]
[ <(-100 : -260 : 1), 1>, <(-56 : 392 : 1), 1>, <(-9 : 78 : 1), 1>,   <(-4 : -28 : 1), 1>, <(0 : 0 : 1), 
1>, <(4 : -52 : 1), 1>, <(56 : -728 : 1), 1> ]

```

And for more edification:


MordellWeilShaInformation(F);


Torsion Subgroup = Z/6
 Analytic rank = 1
 ==> Rank(E) = 1
 The 2-Selmer group has rank 2
 New point of infinite order (x = -56/25)
 After 2-descent: 
 1 <= Rank(E) <= 1
 Sha(E)[2] is trivial
 (Searched up to height 100 on the 2-coverings.)



```
[ 1, 1 ]
[ (-100 : 260 : 1) ]
[
    <2, [ 0, 0 ]>
]

```

Make of this what you will.


