FredH>There are indeed plenty of rational solutions. [...] I never imagined the chord-and-tangent method would work given a crossterm, particularly since there are four real branches instead of the usual one or two. In fact, scaling by 120 sqrt(3) and shifting by 197/100, one gets y^2 = (50 x -197)^2/7500-790/(100 x + 197) which, miraculously, has 120 degree symmetry (as well as up-down). A "sesquihyperbola" surrounding a rounded triangle. And sure enough, no line hits the curve more than three times. Rescaling to whole dollars, the addition and doubling formulas are [a,b0,c0,d0]+[a,b1,c1,d1] = [a,b2,c2,d2] and 2*[a,b,c,d] = [a,b3,c3,d3], with b2=((100*a*(b1-b0)^2*d0*d1)/(100*a*b1*d0*d1+100*a*b0*d0*d1+100*a*b1^2*d1 +100*a^2*b1*d1-711*a*b1*d1+100*a*b0^2*d0+100*a^2*b0*d0-711*a*b0*d0+1422)) 2 b2 = 100 a (b1 - b0) d0 d1/(100 a b1 d0 d1 + 100 a b0 d0 d1 2 2 2 + 100 a b1 d1 + 100 a b1 d1 - 711 a b1 d1 + 100 a b0 d0 2 + 100 a b0 d0 - 711 a b0 d0 + 1422) b3=-((100*a*b^2*(d-c)^2)/(200*a*b^3+100*a^2*b^2-711*a*b^2+711)) 2 2 100 a b (d - c) b3 = - ------------------------------------- 3 2 2 2 200 a b + 100 a b - 711 a b + 711 and cyclically, b <- c <- d <- b for the other coordinates. Assorted solutions: [49/25,2883/1820,169/62,2370/2821],[49/25,4099683/2972866,158623231/171679340,33012030/11595311],[25/26,474/325,169/100,3],[79/105,49/25,25/12,81/35],[49/25,-1805/812,5046/665,-237/1102],[79/25,-88209/50680,32761/5544,-3920/17919],[79/25,-1/20,-5,9],[5/4,-711/8134,-6889/1225,4802/415],[3/2,9604/191,-79/37436,-109443/2450],[79/25,32761/5544,-88209/50680,-3920/17919],[6/5,-6889/955,109443/8300,-1975/31706],[25/8,361/40,-474/95,-24/475],[25/8,-96/25,79/10,-3/40],[25/8,-4050/2821,-75919/273000,66248/11625],[25/8,-4050/2821,66248/11625,-75919/273000],[6/5,-27,395/12,-1/150],[6/5,-147/1300,1975/182,-169/35],[6/5,1975/182,-169/35,-147/1300],[6/5,109443/8300,-6889/955,-1975/31706],[6/5,-881877/257632,1204352/126575,-34445/189344],[49/25,-1805/812,5046/665,-237/1102],[49/25,-64/13715,44521/1456,-600795/23632] some of which came from search results: [1250/13,1896/13,169,300]/100, [1580/21,196,625/3,1620/7]/100, plus the several earlier mentioned. --rwg