2 points, 1 halving line
4 points, 3 halving lines
6 points, 6 halving lines
8 points, 9 halving lines (two different ways)
10 points, 13 halving lines (at least two different ways)
12 points, 18 halving lines

14 points, 22 halving lines:
    (452648907,1343509276)
    (1614815478,365432560)
    (1785603113,994585)
    (19378,1511102143)
    (1270962653,1045917111)
    (1218569805,1002641001)
    (1531100263,3760666810)
    (1489164466,3304054560)
    (1286686514,1147135245)
    (1432238320,771548087)
    (1310730755,1076344722)
    (1327492634,942584266)
    (1054656560,845009118)
    (1298596006,1214967486)

16 points, 27 halving lines: (best I got by hand = 24)
    (1419457110,1288815591)
    (1089703110,2087545580)
    (6677906,207312099)
    (390845440,542231463)
    (1826315345,901217541)
    (1130462073,1645496810)
    (1660230969,981396579)
    (924545120,2705284987)
    (1987972576,768760531)
    (1130438295,1977153458)
    (1131735379,1623563705)
    (1771543299,957197457)
    (1078907170,1246792935)
    (823067128,980581132)
    (2140235774,643603588)
    (1050783094,1039704913)

18 points, 32 halving lines: (best I got by hand = 30)
    (520934855,1806046244)
    (2030783884,1578973986)
    (1085702707,973178912)
    (205501865,2012115372)
    (850282468,1595205536)
    (1614755517,1452534926)
    (1295273780,1378042395)
    (1029964441,626950740)
    (1113812091,1041972412)
    (3524666371,2038058869)
    (46418756,2116077361)
    (750315458,1652192759)
    (932009821,21855284)
    (1077200723,1117189109)
    (1270949952,1334701350)
    (1634522577,1454093161)
    (762409514,1637565112)
    (1160373582,1209938774)