1608:. However, this distance is still short compared to the distance to the object being measured (the height of the triangle) and the skinny triangle solution can be applied and still achieve great accuracy. The alternative method of measuring the base angles is theoretically possible but not so accurate. The base angles are very nearly right angles and would need to be measured with much greater precision than the parallax angle in order to get the same accuracy.
20:
1080:
903:
1611:
The same method of measuring parallax angles and applying the skinny triangle can be used to measure the distances to stars, at least the nearer ones. In the case of stars, however, a longer baseline than the diameter of the Earth is usually required. Instead of using two stations on the baseline,
1725:, relies on making estimates of wind speeds aloft over long distances to calculate a desired heading. Since predicted or reported wind speeds are rarely accurate, corrections to the aircraft's heading need to be made at regular intervals. Skinny triangles form the basis of the
689:
807:
1069:
547:
603:
1678:
in the sight corresponds to a range of 1000 metres. There is an inverse relationship between the angle measured in a sniper's sight and the distance to target. For instance, if this same target measures
1524:
1014:
894:
847:
1584:
Applications of the skinny triangle occur in any situation where the distance to a far object is to be determined. This can occur in surveying, astronomy, and also has military applications.
1654:
The skinny triangle is useful in gunnery in that it allows a relationship to be calculated between the range and size of the target without the shooter needing to compute or look up any
714:
495:
1568:
1473:
1604:
angle formed by the object as seen by the two stations. This baseline is usually very long for best accuracy; in principle the stations could be on opposite sides of the
614:
733:
1636:. There is an inverse relationship between the distance in parsecs and the angle in arcseconds. For instance, two arcseconds corresponds to a distance of
849:
represents the base angle of the triangle and is this value because the sum of the internal angles of any triangle (in this case the two base angles plus
1729:, which is "After travelling 60 miles, your heading is one degree off for every mile you're off course". "60" is very close to 180 / π = 57.30.
1612:
two measurements are made from the same station at different times of year. During the intervening period, the orbit of the Earth around the
1909:
931:
This result is equivalent to assuming that the length of the base of the triangle is equal to the length of the arc of circle of radius
1025:
503:
562:
1713:. Or, perhaps more usefully, a target 6 feet in height and measuring 4 MOA corresponds to a range of 1800 yards (just over a mile).
1957:
1938:
1917:
1892:
1871:
1484:
966:
859:
939:. The error is 10% or less for angles less than about 43°, and improves quadratically: when the angle decreases by a factor of
815:
1982:
1616:
moves the measuring station a great distance, so providing a very long baseline. This baseline can be as long as the
1738:
553:
1655:
1649:
1743:
1596:
objects. The base of the triangle is formed by the distance between two measuring stations and the angle
697:
67:
1694:
as they are with milliradians; however, there is a convenient approximate whole number correspondence in
470:
1977:
43:
1535:
1449:
723:
The proof of the skinny triangle solution follows from the small-angle approximation by applying the
684:{\displaystyle \cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)\approx 1,\quad \theta \ll 1}
71:
802:{\displaystyle {\frac {b}{\sin \theta }}={\frac {r}{\sin \left({\frac {\pi -\theta }{2}}\right)}}}
853:) are equal to π. Applying the small angle approximations to the law of sines above results in
1953:
1934:
1913:
1888:
1867:
1621:
297:
1902:
Holbrow, Charles H.; Lloyd, James N.; Amato, Joseph C.; Galvez, Enrique; Parks, Beth (2010).
1882:
917:
19:
1722:
1695:
910:
464:
The approximated solution to the skinny isosceles triangle, referring to figure 1, is:
63:
1079:
1971:
1691:
1687:
1726:
1690:(MOA). The distances corresponding to minutes of arc are not exact numbers in the
1593:
1443:
The approximated solution to the right skinny triangle, referring to figure 3, is:
724:
31:
1928:
1903:
1861:
1667:
1663:
46:
of such triangles can be greatly simplified by using the approximation that the
1592:
The skinny triangle is frequently used in astronomy to measure the distance to
1633:
1617:
1625:
59:
902:
1601:
1576:
The error of this approximation is less than 10% for angles 31° or less.
953:
58:. The solution is particularly simple for skinny triangles that are also
39:
1659:
77:
The skinny triangle finds uses in surveying, astronomy, and shooting.
1629:
717:
55:
1605:
1078:
901:
51:
1064:{\displaystyle {\text{area}}\approx {\frac {1}{2}}\theta r^{2}\,}
542:{\displaystyle {\text{area}}\approx {\frac {1}{2}}\theta r^{2}\,}
1710:
1701:
1624:(AU). The distance to a star with a parallax angle of only one
957:
598:{\displaystyle \sin \theta \approx \theta ,\quad \theta \ll 1\,}
47:
1860:
Abell, George Ogden; Morrison, David; Wolff, Sidney C. (1987).
1613:
1519:{\displaystyle \tan \theta \approx \theta ,\quad \theta \ll 1}
1640:
and 0.5 arcsecond corresponds to a distance of two parsecs.
1009:{\displaystyle {\text{area}}={\frac {\sin \theta }{2}}r^{2}}
889:{\displaystyle {\frac {b}{\theta }}\approx {\frac {r}{1}}}
1686:
Another unit which is sometimes used on gunsights is the
1628:
measured on a baseline of one AU is a unit known as the
1792:
1658:. Military and hunting telescopic sights often have a
1839:
1837:
1760:
1758:
842:{\displaystyle \scriptstyle {\frac {\pi -\theta }{2}}}
819:
701:
1538:
1487:
1452:
1028:
969:
862:
818:
736:
700:
617:
565:
506:
473:
1019:Applying the small angle approximations results in
1828:
1816:
1804:
1776:
1562:
1518:
1467:
1091:Table of tangent small-angle approximation errors
1063:
1008:
888:
841:
801:
708:
683:
597:
541:
489:
42:whose height is much greater than its base. The
1529:which when substituted into the exact solution
1478:This is based on the small-angle approximation
86:Table of sine small-angle approximation errors
1632:(pc) in astronomy and is equal to about 3.26
8:
1709:in the sight corresponds to a range of 100
1683:in the sight then the range is 500 metres.
1620:of the Earth's orbit or, equivalently, two
1780:
1950:Firearms, the law and forensic ballistics
1537:
1486:
1451:
1060:
1054:
1037:
1029:
1027:
1000:
978:
970:
968:
876:
863:
861:
820:
817:
774:
758:
737:
735:
699:
641:
616:
594:
564:
538:
532:
515:
507:
505:
486:
472:
1744:Infinitesimal oscillations of a pendulum
1272:
1104:
1089:
278:
99:
84:
18:
1754:
1843:
1721:A simple form of aviation navigation,
1666:, in this context usually called just
1910:Springer Science & Business Media
1764:
7:
709:{\displaystyle \scriptstyle \theta }
490:{\displaystyle b\approx r\theta \,}
1829:Abell, Morrison & Wolff (1987)
1817:Abell, Morrison & Wolff (1987)
1805:Abell, Morrison & Wolff (1987)
1777:Abell, Morrison & Wolff (1987)
14:
1563:{\displaystyle b=h\tan \theta \ }
1468:{\displaystyle b\approx h\theta }
74:can be entirely dispensed with.
1506:
727:. Again referring to figure 1:
671:
584:
1930:Basics of Photonics and Optics
66:: in these cases the need for
1:
899:which is the desired result.
1905:Modern Introductory Physics
1863:Exploration of the Universe
1831:, p. 414–416, 418–419.
1573:yields the desired result.
1999:
1884:Physics for Advanced Level
1647:
554:small-angle approximations
54:is equal to that angle in
1739:Small angle approximation
1275:
1107:
1086:The right skinny triangle
943:, the error decreases by
281:
102:
26:Isosceles skinny triangle
1952:. Taylor & Francis.
1881:Breithaupt, Jim (2000).
1866:. Saunders College Pub.
1705:in height and measuring
1674:in height and measuring
1948:Warlow, Tom A. (1996).
1933:. Trafford Publishing.
1670:or mil-dots. A target
1656:trigonometric functions
1650:Telemeter (rangefinder)
954:side-angle-side formula
68:trigonometric functions
1564:
1520:
1469:
1087:
1065:
1010:
928:
890:
843:
803:
710:
685:
599:
543:
491:
27:
1927:Vasan, Srini (2004).
1819:, Inside front cover.
1793:Holbrow et al. (2010)
1565:
1521:
1470:
1082:
1066:
1011:
916:approaches length of
905:
891:
844:
804:
711:
686:
600:
552:This is based on the
544:
492:
22:
1779:, pp. 414–415;
1536:
1485:
1450:
1026:
967:
860:
816:
734:
698:
615:
563:
504:
471:
1092:
960:of the triangle is
935:subtended by angle
87:
1983:Types of triangles
1887:. Nelson Thornes.
1622:astronomical units
1560:
1516:
1465:
1090:
1088:
1061:
1006:
929:
886:
839:
838:
799:
706:
705:
681:
595:
539:
487:
85:
81:Isosceles triangle
28:
1795:, pp. 30–31.
1781:Breithaupt (2000)
1559:
1441:
1440:
1437:
1436:
1269:
1268:
1045:
1032:
994:
973:
884:
871:
836:
797:
790:
753:
649:
523:
510:
462:
461:
458:
457:
275:
274:
1990:
1963:
1944:
1923:
1898:
1877:
1847:
1841:
1832:
1826:
1820:
1814:
1808:
1802:
1796:
1790:
1784:
1774:
1768:
1762:
1708:
1704:
1682:
1677:
1673:
1639:
1569:
1567:
1566:
1561:
1557:
1525:
1523:
1522:
1517:
1474:
1472:
1471:
1466:
1273:
1105:
1093:
1070:
1068:
1067:
1062:
1059:
1058:
1046:
1038:
1033:
1030:
1015:
1013:
1012:
1007:
1005:
1004:
995:
990:
979:
974:
971:
948:
942:
895:
893:
892:
887:
885:
877:
872:
864:
848:
846:
845:
840:
837:
832:
821:
808:
806:
805:
800:
798:
796:
795:
791:
786:
775:
759:
754:
752:
738:
715:
713:
712:
707:
690:
688:
687:
682:
661:
657:
650:
642:
604:
602:
601:
596:
548:
546:
545:
540:
537:
536:
524:
516:
511:
508:
496:
494:
493:
488:
279:
100:
88:
16:Type of triangle
1998:
1997:
1993:
1992:
1991:
1989:
1988:
1987:
1968:
1967:
1966:
1960:
1947:
1941:
1926:
1920:
1901:
1895:
1880:
1874:
1859:
1855:
1850:
1842:
1835:
1827:
1823:
1815:
1811:
1803:
1799:
1791:
1787:
1775:
1771:
1763:
1756:
1752:
1735:
1719:
1706:
1699:
1680:
1675:
1671:
1652:
1646:
1637:
1590:
1582:
1534:
1533:
1483:
1482:
1448:
1447:
1077:
1050:
1024:
1023:
996:
980:
965:
964:
944:
940:
858:
857:
822:
814:
813:
776:
770:
763:
742:
732:
731:
696:
695:
640:
636:
613:
612:
561:
560:
528:
502:
501:
469:
468:
83:
64:right triangles
36:skinny triangle
17:
12:
11:
5:
1996:
1994:
1986:
1985:
1980:
1970:
1969:
1965:
1964:
1958:
1945:
1939:
1924:
1918:
1899:
1893:
1878:
1872:
1856:
1854:
1851:
1849:
1848:
1833:
1821:
1809:
1807:, p. 414.
1797:
1785:
1769:
1767:, p. 124.
1753:
1751:
1748:
1747:
1746:
1741:
1734:
1731:
1723:dead reckoning
1718:
1715:
1696:imperial units
1662:calibrated in
1645:
1642:
1589:
1586:
1581:
1578:
1571:
1570:
1556:
1553:
1550:
1547:
1544:
1541:
1527:
1526:
1515:
1512:
1509:
1505:
1502:
1499:
1496:
1493:
1490:
1476:
1475:
1464:
1461:
1458:
1455:
1439:
1438:
1435:
1434:
1431:
1428:
1424:
1423:
1420:
1417:
1413:
1412:
1409:
1406:
1402:
1401:
1398:
1395:
1391:
1390:
1387:
1384:
1380:
1379:
1376:
1373:
1369:
1368:
1365:
1362:
1358:
1357:
1354:
1351:
1347:
1346:
1343:
1340:
1336:
1335:
1332:
1329:
1325:
1324:
1321:
1318:
1314:
1313:
1310:
1307:
1303:
1302:
1299:
1296:
1292:
1291:
1288:
1285:
1281:
1280:
1277:
1270:
1267:
1266:
1263:
1260:
1256:
1255:
1252:
1249:
1245:
1244:
1241:
1238:
1234:
1233:
1230:
1227:
1223:
1222:
1219:
1216:
1212:
1211:
1208:
1205:
1201:
1200:
1197:
1194:
1190:
1189:
1186:
1183:
1179:
1178:
1175:
1172:
1168:
1167:
1164:
1161:
1157:
1156:
1153:
1150:
1146:
1145:
1142:
1139:
1135:
1134:
1131:
1128:
1124:
1123:
1120:
1117:
1113:
1112:
1109:
1101:
1100:
1097:
1076:
1075:Right triangle
1073:
1072:
1071:
1057:
1053:
1049:
1044:
1041:
1036:
1017:
1016:
1003:
999:
993:
989:
986:
983:
977:
897:
896:
883:
880:
875:
870:
867:
835:
831:
828:
825:
810:
809:
794:
789:
785:
782:
779:
773:
769:
766:
762:
757:
751:
748:
745:
741:
704:
692:
691:
680:
677:
674:
670:
667:
664:
660:
656:
653:
648:
645:
639:
635:
632:
629:
626:
623:
620:
606:
605:
593:
590:
587:
583:
580:
577:
574:
571:
568:
550:
549:
535:
531:
527:
522:
519:
514:
498:
497:
485:
482:
479:
476:
460:
459:
456:
455:
452:
449:
445:
444:
441:
438:
434:
433:
430:
427:
423:
422:
419:
416:
412:
411:
408:
405:
401:
400:
397:
394:
390:
389:
386:
383:
379:
378:
375:
372:
368:
367:
364:
361:
357:
356:
353:
350:
346:
345:
342:
339:
335:
334:
331:
328:
324:
323:
320:
317:
313:
312:
309:
306:
302:
301:
294:
291:
287:
286:
283:
276:
273:
272:
269:
266:
262:
261:
258:
255:
251:
250:
247:
244:
240:
239:
236:
233:
229:
228:
225:
222:
218:
217:
214:
211:
207:
206:
203:
200:
196:
195:
192:
189:
185:
184:
181:
178:
174:
173:
170:
167:
163:
162:
159:
156:
152:
151:
148:
145:
141:
140:
137:
134:
130:
129:
126:
123:
119:
118:
115:
112:
108:
107:
104:
96:
95:
92:
82:
79:
15:
13:
10:
9:
6:
4:
3:
2:
1995:
1984:
1981:
1979:
1976:
1975:
1973:
1961:
1959:0-7484-0432-5
1955:
1951:
1946:
1942:
1940:1-4120-4138-4
1936:
1932:
1931:
1925:
1921:
1919:0-387-79079-9
1915:
1911:
1907:
1906:
1900:
1896:
1894:0-7487-4315-4
1890:
1886:
1885:
1879:
1875:
1873:0-03-005143-6
1869:
1865:
1864:
1858:
1857:
1852:
1846:, p. 87.
1845:
1844:Warlow (1996)
1840:
1838:
1834:
1830:
1825:
1822:
1818:
1813:
1810:
1806:
1801:
1798:
1794:
1789:
1786:
1783:, p. 26.
1782:
1778:
1773:
1770:
1766:
1761:
1759:
1755:
1749:
1745:
1742:
1740:
1737:
1736:
1732:
1730:
1728:
1724:
1716:
1714:
1712:
1703:
1697:
1693:
1692:metric system
1689:
1688:minute of arc
1684:
1669:
1665:
1661:
1657:
1651:
1643:
1641:
1635:
1631:
1627:
1623:
1619:
1615:
1609:
1607:
1603:
1599:
1595:
1587:
1585:
1579:
1577:
1574:
1554:
1551:
1548:
1545:
1542:
1539:
1532:
1531:
1530:
1513:
1510:
1507:
1503:
1500:
1497:
1494:
1491:
1488:
1481:
1480:
1479:
1462:
1459:
1456:
1453:
1446:
1445:
1444:
1432:
1429:
1426:
1425:
1421:
1418:
1415:
1414:
1410:
1407:
1404:
1403:
1399:
1396:
1393:
1392:
1388:
1385:
1382:
1381:
1377:
1374:
1371:
1370:
1366:
1363:
1360:
1359:
1355:
1352:
1349:
1348:
1344:
1341:
1338:
1337:
1333:
1330:
1327:
1326:
1322:
1319:
1316:
1315:
1311:
1308:
1305:
1304:
1300:
1297:
1294:
1293:
1289:
1286:
1283:
1282:
1278:
1274:
1271:
1264:
1261:
1258:
1257:
1253:
1250:
1247:
1246:
1242:
1239:
1236:
1235:
1231:
1228:
1225:
1224:
1220:
1217:
1214:
1213:
1209:
1206:
1203:
1202:
1198:
1195:
1192:
1191:
1187:
1184:
1181:
1180:
1176:
1173:
1170:
1169:
1165:
1162:
1159:
1158:
1154:
1151:
1148:
1147:
1143:
1140:
1137:
1136:
1132:
1129:
1126:
1125:
1121:
1118:
1115:
1114:
1110:
1106:
1103:
1102:
1099:Small angles
1098:
1095:
1094:
1085:
1081:
1074:
1055:
1051:
1047:
1042:
1039:
1034:
1022:
1021:
1020:
1001:
997:
991:
987:
984:
981:
975:
963:
962:
961:
959:
955:
950:
947:
938:
934:
926:
922:
919:
915:
912:
908:
904:
900:
881:
878:
873:
868:
865:
856:
855:
854:
852:
833:
829:
826:
823:
792:
787:
783:
780:
777:
771:
767:
764:
760:
755:
749:
746:
743:
739:
730:
729:
728:
726:
721:
719:
702:
678:
675:
672:
668:
665:
662:
658:
654:
651:
646:
643:
637:
633:
630:
627:
624:
621:
618:
611:
610:
609:
591:
588:
585:
581:
578:
575:
572:
569:
566:
559:
558:
557:
555:
533:
529:
525:
520:
517:
512:
500:
499:
483:
480:
477:
474:
467:
466:
465:
453:
450:
447:
446:
442:
439:
436:
435:
431:
428:
425:
424:
420:
417:
414:
413:
409:
406:
403:
402:
398:
395:
392:
391:
387:
384:
381:
380:
376:
373:
370:
369:
365:
362:
359:
358:
354:
351:
348:
347:
343:
340:
337:
336:
332:
329:
326:
325:
321:
318:
315:
314:
310:
307:
304:
303:
299:
295:
292:
289:
288:
284:
280:
277:
270:
267:
264:
263:
259:
256:
253:
252:
248:
245:
242:
241:
237:
234:
231:
230:
226:
223:
220:
219:
215:
212:
209:
208:
204:
201:
198:
197:
193:
190:
187:
186:
182:
179:
176:
175:
171:
168:
165:
164:
160:
157:
154:
153:
149:
146:
143:
142:
138:
135:
132:
131:
127:
124:
121:
120:
116:
113:
110:
109:
105:
101:
98:
97:
94:Small angles
93:
90:
89:
80:
78:
75:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
25:
21:
1978:Trigonometry
1949:
1929:
1904:
1883:
1862:
1853:Bibliography
1824:
1812:
1800:
1788:
1772:
1765:Vasan (2004)
1727:1 in 60 rule
1720:
1698:. A target
1685:
1664:milliradians
1653:
1610:
1597:
1594:Solar System
1591:
1583:
1580:Applications
1575:
1572:
1528:
1477:
1442:
1096:Large angles
1083:
1018:
951:
945:
936:
932:
930:
924:
920:
913:
906:
898:
850:
811:
725:law of sines
722:
693:
607:
551:
463:
91:Large angles
76:
35:
32:trigonometry
29:
23:
1634:light years
50:of a small
1972:Categories
1750:References
1648:See also:
1618:major axis
909:Length of
1626:arcsecond
1588:Astronomy
1555:θ
1552:
1511:≪
1508:θ
1501:θ
1498:≈
1495:θ
1492:
1463:θ
1457:≈
1287:(radians)
1284:(minutes)
1119:(radians)
1116:(degrees)
1048:θ
1035:≈
988:θ
985:
927:decreases
923:as angle
874:≈
869:θ
830:θ
827:−
824:π
812:The term
784:θ
781:−
778:π
768:
750:θ
747:
703:θ
676:≪
673:θ
663:≈
655:θ
652:−
644:π
634:
625:θ
622:
589:≪
586:θ
579:θ
576:≈
573:θ
570:
526:θ
513:≈
484:θ
478:≈
293:(radians)
290:(minutes)
114:(radians)
111:(degrees)
60:isosceles
1733:See also
1717:Aviation
1602:parallax
1433:−101.54
956:for the
44:solution
40:triangle
1672:1 metre
1660:reticle
1644:Gunnery
1600:is the
1422:−85.32
1411:−70.51
1400:−57.12
1389:−45.13
1378:−34.55
1367:−25.38
1356:−17.63
1345:−11.28
1265:−39.54
1254:−32.78
1243:−26.77
1232:−21.46
1221:−16.80
1210:−12.76
718:radians
56:radians
1956:
1937:
1916:
1891:
1870:
1681:2 mils
1638:0.5 pc
1630:parsec
1558:
1430:0.0175
1419:0.0160
1408:0.0145
1397:0.0131
1386:0.0116
1375:0.0102
1364:0.0087
1353:0.0073
1342:0.0058
1334:−6.35
1331:0.0044
1323:−2.82
1320:0.0029
1312:−0.71
1309:0.0015
1301:−0.03
1298:0.0003
1290:(ppm)
1279:error
1199:−9.31
1188:−6.43
1177:−4.09
1166:−2.30
1155:−1.02
1144:−0.25
1133:−0.01
1111:error
716:is in
454:50.77
451:0.0175
443:42.66
440:0.0160
432:35.26
429:0.0145
421:28.56
418:0.0131
410:22.56
407:0.0116
399:17.28
396:0.0102
388:12.69
385:0.0087
374:0.0073
363:0.0058
352:0.0044
341:0.0029
330:0.0015
319:0.0006
308:0.0003
285:error
271:20.92
260:17.19
249:13.92
238:11.07
128:0.005
106:error
72:tables
24:Fig. 1
1711:yards
1707:1 MOA
1676:1 mil
1606:Earth
1276:angle
1262:1.047
1251:0.960
1240:0.873
1229:0.785
1218:0.698
1207:0.611
1196:0.524
1185:0.436
1174:0.349
1163:0.262
1152:0.175
1141:0.087
1130:0.017
1108:angle
1084:Fig.3
918:chord
907:Fig.2
694:when
377:8.81
366:5.64
355:3.17
344:1.41
333:0.35
322:0.06
311:0.01
282:angle
268:1.047
257:0.960
246:0.873
235:0.785
227:8.61
224:0.698
216:6.50
213:0.611
205:4.72
202:0.524
194:3.25
191:0.436
183:2.06
180:0.349
172:1.15
169:0.262
161:0.51
158:0.175
150:0.13
147:0.087
139:0.02
136:0.035
125:0.017
103:angle
52:angle
38:is a
1954:ISBN
1935:ISBN
1914:ISBN
1889:ISBN
1868:ISBN
1702:inch
1668:mils
1122:(%)
1031:area
972:area
958:area
952:The
608:and
509:area
117:(%)
48:sine
34:, a
1614:Sun
1549:tan
1489:tan
982:sin
911:arc
765:sin
744:sin
631:sin
619:cos
567:sin
298:ppm
70:or
62:or
30:In
1974::
1912:.
1908:.
1836:^
1757:^
1700:1
1427:60
1416:55
1405:50
1394:45
1383:40
1372:35
1361:30
1350:25
1339:20
1328:15
1317:10
1259:60
1248:55
1237:50
1226:45
1215:40
1204:35
1193:30
1182:25
1171:20
1160:15
1149:10
949:.
720:.
556::
448:60
437:55
426:50
415:45
404:40
393:35
382:30
371:25
360:20
349:15
338:10
300:)
265:60
254:55
243:50
232:45
221:40
210:35
199:30
188:25
177:20
166:15
155:10
1962:.
1943:.
1922:.
1897:.
1876:.
1598:θ
1546:h
1543:=
1540:b
1514:1
1504:,
1460:h
1454:b
1306:5
1295:1
1138:5
1127:1
1056:2
1052:r
1043:2
1040:1
1002:2
998:r
992:2
976:=
946:k
941:k
937:θ
933:r
925:θ
921:b
914:l
882:1
879:r
866:b
851:θ
834:2
793:)
788:2
772:(
761:r
756:=
740:b
679:1
669:,
666:1
659:)
647:2
638:(
628:=
592:1
582:,
534:2
530:r
521:2
518:1
481:r
475:b
327:5
316:2
305:1
296:(
144:5
133:2
122:1
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