http://i.imgur.com/YclyNMC.gif Take a unit square. Plot a few thousand points (a,b) that are a rational distance from three vertices (all but (1,1)) in black. Plot the inverse point (a,b)/(a^2+b^2) in red. Can anyone see the ellipse 3 (x^2+ y^2)-4(x+y)+2 x y=0 ? Or the cubic curve 4 (y^3+ x^3)-3 (y^2+ x^2)+2x y ( 2y+ 2x -1)=0 ? --Ed Pegg Jr Points on the ellipse so far: {{-(4/17),16/51}, {28/33,112/99}, {-(140/387),112/129}, {-(224/627),140/209}, {340/627,272/209}, {736/1107,460/369}, {-(476/1539),544/513}, {-(80/1683),28/561}, {868/2211,992/737}, {-(368/1089),3220/3267}, {3280/3363,1148/1121}, {-(372/1441),4960/4323}, {-(1904/5907),1020/1969}, {752/1649,6580/4947}, {588/1921,7840/5763}, {-(1012/2993),5152/8979}, {-(2156/9843),3920/3281}, {-(3008/10659),3948/3553}, {-(448/3569),1540/10707}, {8176/10947,4380/3649}, {3124/12483,5680/4161}, {5056/14691,6636/4897}, {-(1968/5377),12628/16131}, {-(2132/5841),13120/17523}, {-(1820/18867),672/6289}, {-(3692/19491),7952/6497}, {3916/5457,19936/16371}, {6592/7089,22660/21267}, {5044/23859,10864/7953}, {-(2480/8241),11284/24723}, {-(6320/26643),10428/8881}, {9520/34563,15708/11521}, {5424/9073,34804/27219}, {-(12848/36579),11388/12193}, {35620/37587,13152/12529}, {-(5984/37827),2380/12609}, {6188/10001,38080/30003}, {-(1940/11649),43456/34947}, {-(15980/51507),8272/17169}, {-(6052/17441),49840/52323}, {12080/15521,54964/46563}, {2540/13889,56896/41667}, {-(476/57123),160/19041}, {28336/57171,25116/19057}, {-(8636/58179),24384/19393}, {10948/67971,30912/22657}, {-(3808/18737),68068/56211}, {-(20384/70851),9996/23617}, {-(23504/72627),24860/24209}, {67936/74787,27020/24929}, {-(27740/79827),16352/26609}, {-(9044/27537),83776/82611}, {-(12236/84579),4784/28193}, {-(12236/91443),38640/30481}, {-(11536/31553),77044/94659}, {5344/23313,95524/69939}, {73780/96387,38192/32129}, {-(17360/98643),7068/32881}, {15124/105123,47760/35041}, {-(18704/105507),43420/35169}, {45136/106947,47740/35649}, {12988/26961,106960/80883}, {113764/114243,38240/38081}, {-(14420/39729),85696/119187}, {-(14240/41361),74404/124083}, {-(12880/43601),142324/130803}, {-(45440/162963),21868/54321}, {-(19328/54417),149188/163251}, {17708/40913,164032/122739}, {-(63020/172227),46032/57409}, {-(50876/173283),25280/57761}}