CENTRAL LINES

A line in the plane of a triangle is a central line if it passes through two triangle centers. The Encyclopedia of Triangle Centers (ETC) indexes triangle centers as X(1), X(2), X(3), etc. Thus, for example, X(1)X(2) is a central line. In ETC, the entry for X(8) shows that X(8) lies on X(1)X(2) but does not show other centers collinear with those numbered 1, 2 and 8. The list below shows such collinearities.

The first line shown is the line at infinity. Among the centers on this line, the two with least indices in ETC are X(30) and X(511), so that the line is X(30)X(511), or simply 30 511. Similarly, the Euler line, listed as the fifth line, is abbreviated as 2 3.

After the first line, the next few are 1 2, then 1 3, then 1 4, then 2 3, . . . ; generally, for k>2, lines i j such that i + j = k are listed in order of increasing k, with i from 1 up to Floor((k - 1)/2), and skipping lines already listed (e.g., line 4 20 is listed as 2 3.)

You can use this page together with ETC to find collinearities. Suppose you want to find centers collinear with X(316) and X(625). First, in ETC find at X(316) and X(625) the line containing both centers: X(2)X(187). Then on this page, scroll to 2 187 to conclude that X(2), X(187), X(316), X(598), X(625), X(1153), X(1383) are collinear.

For a table of named central lines (e.g., orthic axis, Nagel line, Soddy line) with access to a discussion of each, click MathWorld.

1. LINE 30 511 (THE LINE AT INFINITY:   aα + bβ + cγ = 0)
30 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 674 680 688 690 696 698 700 702 704 706 708 710 712 714 716 718 720 722 724 726 730 732 734 736 740 742 744 746 752 754 758 760 766 768 772 776 778 780 782 784 786 788 790 792 794 796 802 804 806 808 812 814 816 818 824 826 830 832 834 838 888 891 900 912 916 918 924 926 928 952 971 1154 1499 1503 1510 1912 1938 1946
Number of centers on this line: 114

2. LINE 2 3 (EULER LINE:   sin 2A sin(B - C) α + sin 2B sin(C - A) β + sin 2C sin(A - B) γ = 0)
2 3 4 5 20 21 22 23 24 25 26 27 28 29 30 140 186 199 235 237 297 376 377 378 379 381 382 383 384 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 546 547 548 549 550 631 632 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 964 1003 1004 1005 1006 1008 1009 1010 1011 1012 1013 1080 1113 1114 1312 1313 1314 1315 1316 1325 1344 1345 1346 1347 1368 1370 1375 1513 1529 1532 1536 1551 1556 1557 1559 1563 1564 1567 1583 1584 1585 1586 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1628 1650 1651 1656 1657 1658 1816 1817 1883 1884 1885 1889 1894 1904 1906 1907 1981 1982 1984 1985 1995 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2060 2070 2071 2072 2073 2074 2075 2409
Number of centers on this line: 222

3. LINE 3 6 (BROCARD AXIS:   bc(b2 - c2)α + ca(c2 - a2)β + ab(a2 - b2)γ = 0)
3 6 15 16 32 39 50 52 58 61 62 182 187 216 284 371 372 386 389 500 511 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 800 970 991 1030 1151 1152 1160 1161 1182 1192 1207 1326 1333 1340 1341 1342 1343 1350 1351 1379 1380 1384 1504 1505 1570 1578 1579 1609 1620 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1805 1806 1970 1983 2011 2012 2019 2020 2021 2022 2024 2025 2026 2027 2031 2032 2033 2034 2035 2036 2037 2038 2055 2076 2080 2088 2092 2193 2220 2245 2271 2278 2305 2420
Number of centers on this line: 125

4. LINE 230 232 (ORTHIC AXIS:   (cos A)α + (cos B)β + (cos C)γ = 0)
230 231 232 468 523 647 650 676 1637 1886 1990
Number of centers on this line: 11

5. LINE 44 513 (ANTIORTHIC AXIS:   α + β + γ = 0)
44 513 649 650 652 654 656 657 659 661 672 770 798 822 851 896 899 910 1155 1491 1575 1635 1755 2173 2182 2183 2225 2227 2228 2229 2230 2231 2232 2233 2234 2235 2236 2237 2238 2239 2240 2243 2244 2245 2246 2247 2252 2253 2254 2265 2272 2290 2312 2313 2314 2315 2348
Number of centers on this line: 57

6. LINE 187 237 (LEMOINE AXIS:   bcα + caβ + abγ = 0)
187 237 351 512 647 649 663 665 667 669 887 890 902 1055 1495 1960 2223
Number of centers on this line: 17

7. LINE 325 523 (DE LONGCHAMPS LINE:   a3α + b3β + c3γ = 0)
325 523 684 693 850 858 1273 1491
Number of centers on this line: 8

8. LINE 241 514 (GERGONNE LINE:   (1 + cos A)α + (1 + cos B)β + (1 + cos C)γ = 0)
241 514 650 665 905 1323 1375 1465 1638
Number of centers on this line: 9

9. LINE 1 7
(SODDY LINE:   (1+cos A)(cos B - cos C)α + (1+cos B)(cos C - cos A)β + (1+cos C)(cos A - cos B)γ= 0)
1 7 20 77 170 175 176 269 279 347 390 481 482 516 962 990 991 1042 1044 1323 1371 1372 1373 1374 1442 1443 1448 1458 1721 1742 1770 2263 2293
Number of centers on this line: 33

10. LINE 1 2 (NAGEL LINE:   a(b - c)α + b(c - a)β + c(a - b)γ = 0)
1 2 8 10 42 43 78 145 200 239 306 386 387 498 499 519 551 612 614 869 899 936 938 975 976 978 995 997 1026 1103 1125 1149 1189 1193 1198 1201 1210 1644 1647 1698 1714 1722 1737 1961 1998 1999 2000 2057 2340 2398
Number of centers on this line: 50

11. LINE 1 3        (cos B - cos C)α + (cos C - cos A)β + (cos A - cos B)γ = 0
1 3 35 36 40 46 55 56 57 65 165 171 241 260 354 484 517 559 940 942 980 982 986 988 999 1038 1040 1060 1062 1082 1155 1159 1214 1319 1381 1382 1385 1388 1402 1403 1420 1429 1454 1460 1466 1467 1470 1482 1617 1622 1697 1715 1735 1754 1758 1764 1771 1936 2061 2077 2078 2093 2095 2098 2099 2223 2283 2352
Number of centers on this line: 68

12. LINE 1 4        (sec B - sec C)α + (sec C - sec A)β + (sec A - sec B)γ = 0
1 4 33 34 73 223 225 226 243 278 388 497 515 581 944 946 948 950 1056 1058 1064 1066 1068 1070 1072 1079 1457 1478 1479 1490 1519 1528 1541 1543 1558 1659 1699 1745 1750 1785 1838 1848 1870 1877 1891 2356
Number of centers on this line: 46

13. LINE 1 5        (cos(A-B) - cos(A-C))α + (cos(B-C) - cos(B-A))β + (cos(C-A) - cos(C-B)γ = 0)
1 5 11 12 80 119 355 495 496 952 1317 1387 1411 1421 1483 1484 1718 1807 1837 2006
Number of centers on this line: 20

14. LINE 1 6        (b - c)α + (c - a)β + (a - b)γ = 0
1 6 9 37 44 45 72 213 218 219 220 238 392 405 518 611 613 954 956 958 960 984 1001 1023 1083 1100 1104 1107 1108 1124 1191 1203 1212 1279 1335 1386 1449 1453 1616 1642 1713 1723 1724 1728 1736 1743 1757 2087 2176 2256 2257 2273 2284 2300 2323 2324 2329 2427
Number of centers on this line: 58

15. LINE 2 6        a(b2- c2)α + b(c2- a2)β + c(a2- b2)γ = 0
2 6 69 81 86 141 183 193 230 298 299 302 303 323 325 333 343 352 385 391 394 395 396 491 492 524 590 597 599 615 940 965 966 992 1007 1150 1184 1185 1211 1213 1270 1271 1611 1613 1641 1648 1654 1812 1963 1991 1992 1993 1994 2063 2086 2106 2238 2287 2303 2407 2421
Number of centers on this line: 61

16. LINE 2 7        a(b - c)(b + c - a)α + b(c - a)(c + a - b)β + c(a - b)(a + b - c)γ = 0
2 7 9 57 63 142 144 226 307 329 527 553 579 672 894 908 1025 1400 1423 1445 1447 1652 1653 1708 1741 1944 2002 2094 2285 2406
Number of centers on this line: 30

17. LINE 3 7
3 7 943 954
Number of centers on this line: 4

18. LINE 4 6
4 6 53 217 387 393 397 398 1172 1181 1199 1249 1498 1503 1514 1515 1540 1547 1548 1549 1587 1588 1834 1865 1901 1990 2207 2211 2442
Number of centers on this line: 29

19. LINE 3 8
3 8 100 104 944 952 956 1145 1150 1623 1809
Number of centers on this line: 11

20. LINE 4 7
4 7 273 342 938 942 971 1119 1426 1439 1446 1535 1827 1847 1849 1850 1863 1876 1892 1893 1895
Number of centers on this line: 21

21. LINE 5 6
5 6 68 155 485 486 1352 1353 2165
Number of centers on this line: 9

22. LINE 3 9
3 9 84 198 268 282 610 936 965 971 1436 1490 1604 1765 1903 2182
Number of centers on this line: 16

23. LINE 4 8
4 8 72 92 318 321 329 355 517 962 1530 1824 1828 1829 1851 1867 1868 1871 1872 1878 1900 1902
Number of centers on this line: 22

24. LINE 2 11
2 11 55 100 105 120 149 390 497 528 673 1001 1376 1621 2110
Number of centers on this line: 15

25. LINE 3 10
3 10 197 355 515 958 993 1324 1376 1626 2217
Number of centers on this line: 11

26. LINE 4 9
4 9 10 19 40 71 169 242 281 516 573 966 1276 1277 1512 1542 1544 1706 1753 1766 1826 1839 1842 1855 1861 1869 1890 2183 2270 2333 2345 2354
Number of centers on this line: 32

27. LINE 5 8
5 8 1389 1482
Number of centers on this line: 4

28. LINE 6 7
6 7 294 379 651 673 948 1086 1462 1814
Number of centers on this line: 10

29. LINE 2 12
2 12 56 388 529 958 961 1220 1329 1791
Number of centers on this line: 10

30. LINE 3 11
3 11 499 1479
Number of centers on this line: 4

31. LINE 6 8
6 8 594 964 1220 2295 2298 2345
Number of centers on this line: 8

32. LINE 2 13
2 13 16 530 618 622 624 2004
Number of centers on this line: 8

33. LINE 3 12
3 12 498 1478
Number of centers on this line: 4

34. LINE 4 11
4 11 56 104 108 1359 1846
Number of centers on this line: 7

35. LINE 5 10
5 10 517 946 960 970 1329
Number of centers on this line: 7

36. LINE 7 8
7 8 65 69 75 85 319 320 322 377 388 518 883 1122 1231 1441 1463 1469 1909
Number of centers on this line: 19

37. LINE 1 15
1 15 1251 1276 1652 2306
Number of centers on this line: 6

38. LINE 2 14
2 14 15 531 617 619 621 623 2005
Number of centers on this line: 9

39. LINE 4 12
4 12 55 943 1852
Number of centers on this line: 5

40. LINE 6 10
6 10 1377 1378 2049 2051 2214
Number of centers on this line: 7

41. LINE 1 16
1 16 1250 1277 1653
Number of centers on this line: 5

42. LINE 4 13
4 13 61 1523 1524
Number of centers on this line: 5

43. LINE 8 9
8 9 346 390 391 452 728 950 1334 1697 1731 2082 2136 2269 2321 2325 2347
Number of centers on this line: 17

44. LINE 4 14
4 14 62 1522 1525
Number of centers on this line: 5

45. LINE 5 13
5 13 18 62 395 397
Number of centers on this line: 6

46. LINE 7 11
7 11 658 1156 1996
Number of centers on this line: 5

47. LINE 2 17
2 17 62 532 627 629 634 636
Number of centers on this line: 8

48. LINE 4 15
4 15 17 1527 1545
Number of centers on this line: 5

49. LINE 5 14
5 14 17 61 396 398
Number of centers on this line: 6

50. LINE 6 13
6 13 14 113 115 265 381 399 542 1625 1879 1989 2341 2437
Number of centers on this line: 14

51. LINE 7 12
7 12 1268 1788
Number of centers on this line: 4

52. LINE 8 11
8 11 1320 1329 2098
Number of centers on this line: 5

53. LINE 1 19
1 19 28 48 240 284 380 610 1172 1474 1582 1610 1630 1781 1844 1953 1955 1973 2172 2173 2294 2301 2302 2303 2304 2332 2360
Number of centers on this line: 27

54. LINE 2 18
2 18 61 533 628 630 633 635
Number of centers on this line: 8

55. LINE 4 16
4 16 18 1526 1546
Number of centers on this line: 5

56. LINE 2 19
2 19 534 1848 2339
Number of centers on this line: 5

57. LINE 10 11
10 11 121 392 1145 1682
Number of centers on this line: 6

58. LINE 1 21
1 21 31 38 47 58 63 81 191 255 283 595 758 774 846 896 920 968 993 1046 1468 1496 1497 1580 1612 1621 1707 1725 1749 1780 1822 1823 1923 1931 1954 1959 1962 2292 2328
Number of centers on this line: 39

59. LINE 10 12
10 12 65 72 181 209 210 226 442 758 1211 1215
Number of centers on this line: 12

60. LINE 6 17
6 17 18 195 231 233 1209 1506 1656 2439
Number of centers on this line: 10

61. LINE 1 24
1 24 1061 1452
Number of centers on this line: 4

62. LINE 6 19
6 19 34 65 221 478 607 608 1409 1456 1829 1841 1876 1880 2082 2182 2202 2261 2262 2263 2264 2285 2288 2331 2362 2443
Number of centers on this line: 26

63. LINE 1 25
1 25 1036 1039 1763 1773 1829 1905
Number of centers on this line: 8

64. LINE 6 21
6 21 941 1778
Number of centers on this line: 4

65. LINE 6 22
6 22 251 1176 1180 1501
Number of centers on this line: 6

66. LINE 7 21
7 21 56 86 348 1001 1014 1284 1434 1444 1804
Number of centers on this line: 11

67. LINE 8 20
8 20 40 63 84 189 271 280 515 1158 1782 2357
Number of centers on this line: 12

68. LINE 13 15
13 15 30 396
Number of centers on this line: 4

69. LINE 6 23
6 23 353 1383
Number of centers on this line: 4

70. LINE 8 21
8 21 55 333 345 958 1043 1259 1792
Number of centers on this line: 9

71. LINE 1 29
1 29 92 158 820 1047 1096 1784 1895 1896
Number of centers on this line: 10

72. LINE 6 24
6 24 54 973
Number of centers on this line: 4

73. LINE 9 21
9 21 41 78 284 1819 2268 2287 2289 2327
Number of centers on this line: 10

74. LINE 14 16
14 16 30 395
Number of centers on this line: 4

75. LINE 1 30
1 30 79 500 554 1081 1464 1717 1836
Number of centers on this line: 9

76. LINE 6 25
6 25 51 154 159 161 184 206 232 1194 1474 1495 1660 1843 1915 1971 1974 2194 2203 2299 2445
Number of centers on this line: 21

77. LINE 10 21
10 21 35 80 100 759 1283 1793
Number of centers on this line: 8

78. LINE 1 32
1 32 172 609 760 985 1572 1759 1914 2241 2242 2243 2344
Number of centers on this line: 13

79. LINE 2 31
2 31 171 238 748 750 752 985 2239
Number of centers on this line: 9

80. LINE 15 18
15 18 140 398
Number of centers on this line: 4

81. LINE 16 17
16 17 140 397
Number of centers on this line: 4

82. LINE 2 32
2 32 83 251 315 626 754 1078 1369 1506 1627 1799 2240
Number of centers on this line: 13

83. LINE 3 31
3 31 601 602 1036 1064 1193 1245 1472 2221
Number of centers on this line: 10

84. LINE 7 27
7 27 81 222 278 608 1172 1396
Number of centers on this line: 8

85. LINE 2 33
2 33 1040 1861
Number of centers on this line: 4

86. LINE 3 33
3 33 1753 1872
Number of centers on this line: 4

87. LINE 4 32
4 32 98 112 115 132 248 1561 1968
Number of centers on this line: 9

88. LINE 6 31
6 31 42 55 71 209 212 605 606 672 674 902 1011 1202 1206 1250 1253 1914 1918 2066 2177 2209 2266 2267 2268 2269 2276 2280 2293 2308 2309 2330 2361 2426
Number of centers on this line: 34

89. LINE 8 29
8 29 219 281 607 1172 2287 2322
Number of centers on this line: 8

90. LINE 2 36
2 36 535 993 1478
Number of centers on this line: 5

91. LINE 2 37
2 37 75 192 312 321 344 345 346 350 536 1229 1267 1278 1575 1646 1997 2276 2277 2345 2397
Number of centers on this line: 21

92. LINE 6 33
6 33 204 1864 1903 2192
Number of centers on this line: 6

93. LINE 10 29
10 29 1794 2328
Number of centers on this line: 4

94. LINE 1 39
1 39 291 292 1009 1015 1018 1500 2007 2008 2108 2275 2276 2295
Number of centers on this line: 14

95. LINE 2 38
2 38 244 291 335 537 756 982 984 1215
Number of centers on this line: 10

96. LINE 3 37
3 37 975 990 1766 2178
Number of centers on this line: 6

97. LINE 9 31
9 31 612 756 2298
Number of centers on this line: 5

98. LINE 2 39
2 39 76 194 274 305 310 538 980 1180 1194 1196 1228 1230 1645 1655 2229 2396
Number of centers on this line: 18

99. LINE 4 37
4 37 1841 2335
Number of centers on this line: 4

100. LINE 7 34
7 34 1039 1890
Number of centers on this line: 4

101. LINE 10 31
10 31 964 1724
Number of centers on this line: 4

102. LINE 1 41
1 41 101 105 169 294 1183 1282 1438 1752 2082 2112 2170 2246 2280
Number of centers on this line: 15

103. LINE 2 40
2 40 946 962
Number of centers on this line: 4

104. LINE 6 36
6 36 609 2163 2251 2364
Number of centers on this line: 6

105. LINE 8 34
8 34 1041 1861
Number of centers on this line: 4

106. LINE 9 33
9 33 212 1172 2212 2299 2328
Number of centers on this line: 7

107. LINE 4 39
4 39 232 262 1555
Number of centers on this line: 5

108. LINE 3 41
3 41 218 672 949
Number of centers on this line: 5

109. LINE 5 39
5 39 114 115 1506 1569 2009 2010 2023
Number of centers on this line: 9

110. LINE 7 37
7 37 241 1418
Number of centers on this line: 4

111. LINE 9 35
9 35 90 2164
Number of centers on this line: 4

112. LINE 11 33
11 33 427 614 1862
Number of centers on this line: 5

113. LINE 19 25
19 25 33 37 55 197 198 228 612 910 968 1486 1696 1824 1827 1859 2201 2355
Number of centers on this line: 18

114. LINE 3 42
3 42 967 1468
Number of centers on this line: 4

115. LINE 5 40
5 40 1698 1699
Number of centers on this line: 4

116. LINE 8 37
8 37 941 966
Number of centers on this line: 4

117. LINE 2 44
2 44 89 320
Number of centers on this line: 4

118. LINE 6 40
6 40 380 937 1702 1703 2255
Number of centers on this line: 7

119. LINE 12 34
12 34 427 612
Number of centers on this line: 4

120. LINE 19 27
19 27 63 75 92 286 333 1748 1755 1760 1761 1762 1965 2322
Number of centers on this line: 14

121. LINE 2 45
2 45 88 190 545 903 1086
Number of centers on this line: 7

122. LINE 6 41
6 41 48 56 73 172 198 604 1055 1193 1400 1404 1405 1428 1450 1451 1458 1468 1469 1471 1475 1951 2067 2174 2178 2183 2260 2274 2275 2277 2286 2307 2317 2347 2425
Number of centers on this line: 35

123. LINE 9 38
9 38 614 748
Number of centers on this line: 4

124. LINE 10 37
10 37 594 740 1187 1213 1500 1834 2092 2321
Number of centers on this line: 10

125. LINE 9 39
9 39 978 988 992 2277
Number of centers on this line: 6

126. LINE 10 38
10 38 596 1739
Number of centers on this line: 4

127. LINE 6 43
6 43 87 171 1197 1376 1575 1740 2162 2319
Number of centers on this line: 10

128. LINE 8 41
8 41 2329 2344
Number of centers on this line: 4

129. LINE 10 39
10 39 730 1107 1573 1574 1575 2013 2014
Number of centers on this line: 9

130. LINE 12 37
12 37 225 227 429 1826 1832 1833 1840 1865 1867 1873 1874 1880 1881 1882 1893 2197
Number of centers on this line: 18

131. LINE 3 47
3 47 1399 2361
Number of centers on this line: 4

132. LINE 4 46
4 46 90 920 1158 1452 1708 1709 1727 1728 1737 1770 1774 1775 1776 1779 1788 1940
Number of centers on this line: 18

133. LINE 15 35
15 35 1250 2307
Number of centers on this line: 4

134. LINE 19 31
19 31 204 1096 2181 2299 2312
Number of centers on this line: 7

135. LINE 3 48
3 48 71 219 916 1802 1818 2200 2252 2289
Number of centers on this line: 10

136. LINE 9 42
9 42 941 968 2258
Number of centers on this line: 5

137. LINE 15 36
15 36 202 1094
Number of centers on this line: 4

138. LINE 3 49
3 49 155 184 185 283 394 1092 1147 1181 1204 1216 1437 1790 1800 1801 1819
Number of centers on this line: 17

139. LINE 9 43
9 43 256 846 893 1045 1711 1716 2092 2238 2276
Number of centers on this line: 11

140. LINE 16 36
16 36 203 1095
Number of centers on this line: 4

141. LINE 2 51
2 51 262 263 373 511
Number of centers on this line: 6

142. LINE 5 49
5 49 54 110 265 567 1141
Number of centers on this line: 7

143. LINE 4 51
4 51 185 389 1075 1093 1896 1899 2052
Number of centers on this line: 9

144. LINE 9 46
9 46 79 191 442 1213 1329 1698 1744 1756 1759 1761 1781 1901 2160 2245
Number of centers on this line: 16

145. LINE 2 54
2 54 68 96 539 569 1147 1209
Number of centers on this line: 8

146. LINE 4 52
4 52 68 324 847
Number of centers on this line: 5

147. LINE 5 51
5 51 52 143 343 973 1154 1209 1568
Number of centers on this line: 9

148. LINE 10 46
10 46 63 377 1478 1773
Number of centers on this line: 6

149. LINE 25 31
25 31 608 1395 1400 1460 2199 2212 2354
Number of centers on this line: 9

150. LINE 3 54
3 54 97 195 1154 1157 1493 1993
Number of centers on this line: 8

151. LINE 9 48
9 48 101 104 214 572 909 993 997 1006 2250 2261 2265 2267
Number of centers on this line: 14

152. LINE 21 36
21 36 79 229 1125 1789
Number of centers on this line: 6

153. LINE 25 32
25 32 1184 1196 2204 2207
Number of centers on this line: 6

154. LINE 4 54
4 54 184 275 578 1614 1629
Number of centers on this line: 7

155. LINE 5 53
5 53 216 233
Number of centers on this line: 4

156. LINE 21 37
21 37 172 1169 1333 1791 2298 2303 2363
Number of centers on this line: 9

157. LINE 25 34
25 34 56 204 208 614 1035 1104 1398 1410 1426 1427 1430 1435 1455 1828 1875
Number of centers on this line: 17

158. LINE 2 58
2 58 540 1330 1724
Number of centers on this line: 5

159. LINE 5 55
5 55 498 1479
Number of centers on this line: 4

160. LINE 19 41
19 41 1825 2171 2259
Number of centers on this line: 5

161. LINE 1 60
1 60 110 245 759
Number of centers on this line: 5

162. LINE 4 57
4 57 84 208 1210 1712 1856 2096
Number of centers on this line: 8

163. LINE 5 56
5 56 499 1478
Number of centers on this line: 4

164. LINE 1 61
1 61 203 2307
Number of centers on this line: 4

165. LINE 28 34
28 34 57 58 603 1393 1394 1395 1396 1448 1451 1452 1453 1835 1866 2299
Number of centers on this line: 16

166. LINE 29 33
29 33 78 312 318 1043
Number of centers on this line: 6

167. LINE 6 57
6 57 222 223 269 610 910 1190 1407 1418 1419 1427 1439 1465 1615 1730 1763 2003 2097 2270
Number of centers on this line: 20

168. LINE 19 44
19 44 1828 1888
Number of centers on this line: 4

169. LINE 29 34
29 34 77 85 86 273 286 1874
Number of centers on this line: 8

170. LINE 31 32
31 32 41 213 228 766 869 1185 1188 1197 2198 2205 2210 2223 2225 2251 2304
Number of centers on this line: 17

171. LINE 8 56
8 56 404 1222 1261 1376 1476 1788
Number of centers on this line: 8

172. LINE 9 55
9 55 200 210 380 480 1260 1864 2264 2348
Number of centers on this line: 10

173. LINE 28 36
28 36 1838 1842
Number of centers on this line: 4

174. LINE 31 33
31 33 2250 2310 2342
Number of centers on this line: 5

175. LINE 8 57
8 57 1219 1706
Number of centers on this line: 4

176. LINE 9 56
9 56 1696 2297
Number of centers on this line: 4

177. LINE 10 55
10 55 405 950 1751 1837 2218
Number of centers on this line: 7

178. LINE 19 46
19 46 579 1714 1723 1731 1732 1738 1752 1782 1838 2252
Number of centers on this line: 12

179. LINE 3 63
3 63 72 78 224 228 912 1259 1260 1473
Number of centers on this line: 10

180. LINE 6 60
6 60 1169 1187 1798
Number of centers on this line: 5

181. LINE 10 56
10 56 474 956 996
Number of centers on this line: 5

182. LINE 19 47
19 47 921 2314
Number of centers on this line: 4

183. LINE 25 41
25 41 42 205 607 1973 2187 2200 2271 2333 2356
Number of centers on this line: 11

184. LINE 31 35
31 35 386 1203
Number of centers on this line: 4

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484. LINE 24 110
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Number of centers on this line: 5

485. LINE 30 104
30 104 1290 1484
Number of centers on this line: 4

486. LINE 56 78
56 78 480 518 1037 1445
Number of centers on this line: 6

487. LINE 57 77
57 77 81 284 604 1014 1412 1449 1786 1790 1817 2280
Number of centers on this line: 12

488. LINE 4 131
4 131 135 925 1299
Number of centers on this line: 5

489. LINE 25 110
25 110 1112 1351 1993
Number of centers on this line: 5

490. LINE 57 78
57 78 404 951 1257
Number of centers on this line: 5

491. LINE 64 71
64 71 198 220 2155
Number of centers on this line: 5

492. LINE 24 112
24 112 2079 2207
Number of centers on this line: 4

493. LINE 25 111
25 111 112 1384 1627
Number of centers on this line: 5

494. LINE 28 108
28 108 225 759 2006
Number of centers on this line: 5

495. LINE 36 101
36 101 672 840 1055
Number of centers on this line: 5

496. LINE 37 100
37 100 111 897
Number of centers on this line: 4

497. LINE 37 101
37 101 284 759 1807 2161 2174 2341
Number of centers on this line: 8

498. LINE 47 91
47 91 92 564 2190
Number of centers on this line: 5

499. LINE 34 106
34 106 108 1420 1421
Number of centers on this line: 5

500. LINE 48 92
48 92 2167 2313
Number of centers on this line: 4

501. LINE 63 77
63 77 219 222 394 1214 1409 2286
Number of centers on this line: 8

502. LINE 4 137
4 137 933 1141
Number of centers on this line: 4

503. LINE 31 110
31 110 593 741
Number of centers on this line: 4

504. LINE 40 101
40 101 972 1802
Number of centers on this line: 4

505. LINE 56 85
56 85 870 1447
Number of centers on this line: 4

506. LINE 67 74
67 74 935 1503
Number of centers on this line: 4

507. LINE 69 72
69 72 304 1264 1265
Number of centers on this line: 5

508. LINE 36 106
36 106 901 902 1149 1318 2226
Number of centers on this line: 7

509. LINE 49 93
49 93 94 565
Number of centers on this line: 4

510. LINE 57 85
57 85 274 333 1424 1434
Number of centers on this line: 6

511. LINE 58 84
58 84 990 1172 2257
Number of centers on this line: 5

512. LINE 69 73
69 73 77 78 307 326 348 1038 1818
Number of centers on this line: 9

513. LINE 1 142
1 142 277 443 1738 1818 2191
Number of centers on this line: 7

514. LINE 11 132
11 132 1848 1859
Number of centers on this line: 4

515. LINE 30 113
30 113 1495 1511 1514 1524 1525 1531 1533 1539 1544 1545 1546 1553 1554 1555 1558 1561 1568
Number of centers on this line: 19

516. LINE 42 101
42 101 111 923
Number of centers on this line: 4

517. LINE 56 87
56 87 238 1423 1424 1463 1935
Number of centers on this line: 7

518. LINE 69 74
69 74 99 376 542 617 1272 1494
Number of centers on this line: 8

519. LINE 58 86
58 86 238 261 320 757 1098 1125 1509
Number of centers on this line: 9

520. LINE 65 79
65 79 80 265 2166
Number of centers on this line: 5

521. LINE 71 73
71 73 1409 2197
Number of centers on this line: 4

522. LINE 3 142
3 142 516 946 1001 1125 1486 1631 2140
Number of centers on this line: 9

523. LINE 30 115
30 115 187 230
Number of centers on this line: 4

524. LINE 36 109
36 109 953 1457
Number of centers on this line: 4

525. LINE 57 88
57 88 651 1405 1787 1797 2246 2316
Number of centers on this line: 8

526. LINE 71 74
71 74 101 2159
Number of centers on this line: 4

527. LINE 72 73
72 73 201 1214 2318
Number of centers on this line: 5

528. LINE 5 141
5 141 211 511 623 624 625 626 635 636 639 640 1216 2039 2040
Number of centers on this line: 15

529. LINE 57 89
57 89 1404 1443 2364
Number of centers on this line: 5

530. LINE 65 81
65 81 961 1408 1798 2363
Number of centers on this line: 6

531. LINE 72 74
72 74 100 2349
Number of centers on this line: 4

532. LINE 55 92
55 92 243 1013
Number of centers on this line: 4

533. LINE 65 82
65 82 1176 1386 1428
Number of centers on this line: 5

534. LINE 4 145
4 145 149 153 952 1320 1482 1537 1862 1897
Number of centers on this line: 10

535. LINE 40 109
40 109 255 1295 1394
Number of centers on this line: 5

536. LINE 10 140
10 140 214 952 1385
Number of centers on this line: 5

537. LINE 4 147
4 147 148 194 1916
Number of centers on this line: 5

538. LINE 6 145
6 145 346 1222
Number of centers on this line: 4

539. LINE 10 141
10 141 142 495 518 942
Number of centers on this line: 6

540. LINE 4 150
4 150 152 331
Number of centers on this line: 4

541. LINE 2 154
2 154 1503 1853
Number of centers on this line: 4

542. LINE 55 101
55 101 2291 2301
Number of centers on this line: 4

543. LINE 56 100
56 100 145 1120 1317 1811
Number of centers on this line: 6

544. LINE 25 132
25 132 136 1624
Number of centers on this line: 4

545. LINE 56 101
56 101 218 1362 1477
Number of centers on this line: 5

546. LINE 57 100
57 100 1280 1477 1810
Number of centers on this line: 5

547. LINE 51 107
51 107 275 436 1298
Number of centers on this line: 5

548. LINE 55 103
55 103 109 222 991 1362 1364
Number of centers on this line: 7

549. LINE 56 102
56 102 1364 1433
Number of centers on this line: 4

550. LINE 4 155
4 155 254 1941 1993
Number of centers on this line: 5

551. LINE 55 104
55 104 1000 1317
Number of centers on this line: 4

552. LINE 58 101
58 101 172 213 292 295 741 1922 2311
Number of centers on this line: 9

553. LINE 3 158
3 158 243 1940
Number of centers on this line: 4

554. LINE 5 156
5 156 184 569
Number of centers on this line: 4

555. LINE 56 105
56 105 279 934 1358 1360
Number of centers on this line: 6

556. LINE 58 103
58 103 112 2332
Number of centers on this line: 4

557. LINE 76 85
76 85 226 304 312 349 1233 1920 2064
Number of centers on this line: 9

558. LINE 50 112
50 112 477 1990
Number of centers on this line: 4

559. LINE 56 106
56 106 109 595 1357 1361 1417
Number of centers on this line: 7

560. LINE 6 157
6 157 248 1576 1976
Number of centers on this line: 5

561. LINE 55 108
55 108 196 972 1360
Number of centers on this line: 5

562. LINE 63 100
63 100 103 165 200 1282 1768
Number of centers on this line: 7

563. LINE 1 163
1 163 293 1910 2083 2247
Number of centers on this line: 6

564. LINE 3 161
3 161 1209 1853
Number of centers on this line: 4

565. LINE 58 106
58 106 110 1201 1797
Number of centers on this line: 5

566. LINE 1 164
1 164 258 266 1130
Number of centers on this line: 5

567. LINE 20 145
20 145 517 944 2096
Number of centers on this line: 5

568. LINE 52 113
52 113 135 235 1112
Number of centers on this line: 5

569. LINE 53 112
53 112 1141 1989
Number of centers on this line: 4

570. LINE 6 160
6 160 237 263
Number of centers on this line: 4

571. LINE 21 145
21 145 956 1000
Number of centers on this line: 4

572. LINE 54 112
54 112 217 1298 1970 1987
Number of centers on this line: 6

573. LINE 2 165
2 165 516 1699
Number of centers on this line: 4

574. LINE 1 167
1 167 174 177 503 2089
Number of centers on this line: 6

575. LINE 6 162
6 162 1013 2326
Number of centers on this line: 4

576. LINE 10 158
10 158 318 1785 2052
Number of centers on this line: 5

577. LINE 53 115
53 115 133 800 1596
Number of centers on this line: 5

578. LINE 1 168
1 168 173 259 1129
Number of centers on this line: 5

579. LINE 71 100
71 100 2249 2287
Number of centers on this line: 4

580. LINE 76 95
76 95 96 1225
Number of centers on this line: 4

581. LINE 3 169
3 169 910 1212
Number of centers on this line: 4

582. LINE 74 98
74 98 690 879 2394
Number of centers on this line: 5

583. LINE 9 165
9 165 910 1376 1709 1750
Number of centers on this line: 6

584. LINE 75 99
75 99 261 759 993
Number of centers on this line: 5

585. LINE 4 171
4 171 601 987 1395
Number of centers on this line: 5

586. LINE 2 174
2 174 236 2090
Number of centers on this line: 4

587. LINE 30 146
30 146 323 399 1138
Number of centers on this line: 5

588. LINE 51 125
51 125 132 427 1112
Number of centers on this line: 5

589. LINE 19 158
19 158 1712 2083 2179
Number of centers on this line: 5

590. LINE 67 110
67 110 141 1176
Number of centers on this line: 4

591. LINE 85 92
85 92 331 1847 1855
Number of centers on this line: 5

592. LINE 2 176
2 176 1659 2362
Number of centers on this line: 4

593. LINE 24 154
24 154 1181 1614
Number of centers on this line: 4

594. LINE 22 157
22 157 183 311
Number of centers on this line: 4

595. LINE 69 110
69 110 206 1177 2373
Number of centers on this line: 5

596. LINE 78 101
78 101 205 2172
Number of centers on this line: 4

597. LINE 2 178
2 178 188 234 557 558 2089
Number of centers on this line: 7

598. LINE 26 154
26 154 155 156 1154
Number of centers on this line: 5

599. LINE 19 162
19 162 897 2247
Number of centers on this line: 4

600. LINE 1 181
1 181 970 1682
Number of centers on this line: 4

601. LINE 9 173
9 173 177 188 236
Number of centers on this line: 5

602. LINE 22 160
22 160 325 1485
Number of centers on this line: 4

603. LINE 1 182
1 182 983 1428 1672 1673 1674 1675 2330
Number of centers on this line: 9

604. LINE 12 171
12 171 1399 1935
Number of centers on this line: 4

605. LINE 22 161
22 161 343 1503
Number of centers on this line: 4

606. LINE 8 177
8 177 556 1143 1274
Number of centers on this line: 5

607. LINE 86 99
86 99 106 551 903
Number of centers on this line: 5

608. LINE 1 185
1 185 296 1020 1425 1937
Number of centers on this line: 6

609. LINE 8 178
8 178 236 483
Number of centers on this line: 4

610. LINE 30 156
30 156 1147 1660
Number of centers on this line: 4

611. LINE 81 105
81 105 110 354 1386 1814 2194
Number of centers on this line: 7

612. LINE 5 182
5 182 206 1348 1349 1503 1676 1677
Number of centers on this line: 8

613. LINE 6 181
6 181 197 1397 1460
Number of centers on this line: 5

614. LINE 1 188
1 188 361 1489
Number of centers on this line: 4

615. LINE 2 187
2 187 316 598 625 1153 1383
Number of centers on this line: 7

616. LINE 27 162
27 162 673 1890 2299
Number of centers on this line: 5

617. LINE 94 96
94 96 925 1141
Number of centers on this line: 4

618. LINE 66 125
66 125 1177 1974
Number of centers on this line: 4

619. LINE 76 115
76 115 626 1916
Number of centers on this line: 4

620. LINE 10 182
10 182 1678 1679 1680 1681
Number of centers on this line: 6

621. LINE 49 143
49 143 1493 1994
Number of centers on this line: 4

622. LINE 87 106
87 106 932 993
Number of centers on this line: 4

623. LINE 88 105
88 105 901 1155
Number of centers on this line: 4

624. LINE 53 141
53 141 264 297 327
Number of centers on this line: 5

625. LINE 95 99
95 99 311 328 1141
Number of centers on this line: 5

626. LINE 37 158
37 158 281 393 406 1123 1336
Number of centers on this line: 7

627. LINE 55 140
55 140 496 499
Number of centers on this line: 4

628. LINE 56 140
56 140 495 498
Number of centers on this line: 4

629. LINE 8 190
8 190 528 1121 1156
Number of centers on this line: 5

630. LINE 98 100
98 100 228 321 1281 1821
Number of centers on this line: 6

631. LINE 1 198
1 198 2262 2270
Number of centers on this line: 4

632. LINE 4 195
4 195 399 1263
Number of centers on this line: 4

633. LINE 7 192
7 192 335 2171
Number of centers on this line: 4

634. LINE 99 100
99 100 668 799
Number of centers on this line: 4

635. LINE 6 194
6 194 384 698 1975
Number of centers on this line: 5

636. LINE 8 192
8 192 256 257 585 586 740 984 1654 1655 2292
Number of centers on this line: 11

637. LINE 10 190
10 190 671 897
Number of centers on this line: 4

638. LINE 99 101
99 101 190 645 662
Number of centers on this line: 5

First edition of this list: September 18, 2003
Second: 12/10/03

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