#define the elliptic curve corresponding to the equation a/(b+c)+b/(a+c)+c/(a+b)=4
ee=EllipticCurve([0,109,0,224,0])

ee

ee.rank()

ee.gens()

P=ee(-100,260) # this is a generator for the group of rational points

2*P # just a check

# Convert a point (x,y) on the elliptic curve back to (a,b,c). This has N as an argument, but in our case N=4.
# We also ensure that the result is all integers, by multiplying by a suitable lcm.
def orig(P,N):
    x=P[0]
    y=P[1]
    a=(8*(N+3)-x+y)/(2*(N+3)*(4-x))
    b=(8*(N+3)-x-y)/(2*(N+3)*(4-x))
    c=(-4*(N+3)-(N+2)*x)/((N+3)*(4-x))
    da=denominator(a)
    db=denominator(b)
    dc=denominator(c)
    l=lcm(da,lcm(db,dc))
    return [a*l,b*l,c*l]

orig(P,4)

u=orig(9*P,4) # Bremner and MacLeod noticed that 9P yields a positive solution. 

(a,b,c)=(u[0],u[1],u[2])
(a,b,c)

a/(b+c)+b/(a+c)+c/(a+b)

a%100003