39:
1642:
31:
1673:
In 2015, an anonymous
Japanese woman using the pen name "aerile re" published the first known method (the method of 3 circumcenters) to construct a proof in elementary geometry for a special class of adventitious quadrangles problem. This work solves the first of the three unsolved problems listed by
1665:
in 1936 (Beantwoording van prijsvraag # 17, Nieuw-Archief voor
Wiskunde 18, pages 14–66). He in fact classified (though with a few errors) all multiple intersections of diagonals in regular polygons. His results (all done by hand) were confirmed with computer, and the errors corrected, by Bjorn
1657:
when measured in degrees or other units for which the whole circle is a rational number. Numerous adventitious quadrangles beyond the one appearing in
Langley's puzzle have been constructed. They form several infinite families and an additional set of sporadic examples.
583:
in 1923. This solution involves drawing one additional line, and then making repeated use of the fact that the internal angles of a triangle add up to 180° to prove that several triangles drawn within the large triangle are all isosceles.
164:
1661:
Classifying the adventitious quadrangles (which need not be convex) turns out to be equivalent to classifying all triple intersections of diagonals in regular polygons. This was solved by
1626:
1571:
1300:
471:
1388:
1344:
1185:
1072:
1028:
984:
871:
827:
783:
1633:
Many other solutions are possible. Cut the Knot list twelve different solutions and several alternative problems with the same 80-80-20 triangle but different internal angles.
547:
512:
638:
574:
347:
221:
1449:
1133:
932:
1498:
1217:
1524:
1414:
1243:
1098:
897:
730:
96:
684:
661:
611:
416:
393:
370:
320:
290:
267:
244:
194:
704:
1738:
1940:
105:
1666:
Poonen and
Michael Rubinstein in 1998. The article contains a history of the problem and a picture featuring the regular
580:
2001:
1823:
1756:
1699:
1584:
1529:
1261:
429:
55:
1349:
1305:
1146:
1033:
989:
945:
832:
788:
744:
49:
is a puzzle in which one must infer an angle in a geometric diagram from other given angles. It was posed by
1791:
1694:
515:
50:
521:
486:
1890:
1840:
1773:
616:
552:
325:
199:
99:
1962:
38:
1921:
1734:
1728:
1795:
17:
1882:
1832:
1765:
1724:
1419:
1103:
902:
1852:
1641:
1471:
1190:
1848:
1654:
1653:
when the angles between its diagonals and sides are all rational angles, angles that give
1503:
1393:
1222:
1077:
876:
709:
75:
666:
643:
593:
398:
375:
352:
302:
272:
249:
226:
176:
689:
1985:
1909:
1995:
1946:
1894:
1667:
30:
1867:
1246:
576:, but being of only finite precision, always leave doubt about the exact value.
1910:"The adventitious quadrangles was solved completely by the elementary solution"
1886:
1868:"The number of intersection points made by the diagonals of a regular polygon"
1662:
1925:
1821:
Rigby, J. F. (1978), "Adventitious quadrangles: a geometrical approach",
1844:
1777:
1730:
The
Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes
1836:
1769:
1640:
37:
579:
A direct proof using classical geometry was developed by
159:{\displaystyle \angle {CBA}=\angle {ACB}=80^{\circ }.}
1587:
1532:
1506:
1474:
1422:
1396:
1352:
1308:
1264:
1225:
1193:
1149:
1106:
1080:
1036:
992:
948:
905:
879:
835:
791:
747:
712:
692:
669:
646:
619:
596:
555:
524:
489:
432:
401:
378:
355:
328:
305:
275:
252:
229:
202:
179:
108:
78:
1459:
Therefore all the red lines in the figure are equal.
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1443:
1408:
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821:
777:
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465:
410:
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261:
238:
215:
188:
158:
90:
67:In its original form the problem was as follows:
42:Solution to Langley's 80-80-20 triangle problem
8:
1754:Tripp, Colin (1975), "Adventitious angles",
1866:Poonen, Bjorn; Rubinstein, Michael (1998),
1968:- English translation of the article from
1942:The last challenge from Geometry the Great
1649:A quadrilateral such as BCEF is called an
1719:
1717:
1715:
1621:{\displaystyle \angle {BEF}=30^{\circ }.}
1609:
1591:
1586:
1566:{\displaystyle \angle {GEF}=70^{\circ }.}
1554:
1536:
1531:
1505:
1473:
1421:
1395:
1374:
1356:
1351:
1330:
1312:
1307:
1295:{\displaystyle \angle {BGE}=100^{\circ }}
1286:
1268:
1263:
1224:
1192:
1171:
1153:
1148:
1105:
1079:
1058:
1040:
1035:
1014:
996:
991:
970:
952:
947:
904:
878:
857:
839:
834:
813:
795:
790:
769:
751:
746:
711:
691:
668:
645:
624:
618:
595:
560:
554:
528:
523:
493:
488:
466:{\displaystyle \angle {BEF}=30^{\circ }.}
454:
436:
431:
400:
377:
354:
333:
327:
304:
274:
251:
228:
207:
201:
178:
147:
129:
112:
107:
77:
1383:{\displaystyle \angle {GEB}=40^{\circ }}
1339:{\displaystyle \angle {GBE}=40^{\circ }}
1180:{\displaystyle \angle {FBG}=60^{\circ }}
1067:{\displaystyle \angle {BFC}=50^{\circ }}
1023:{\displaystyle \angle {CBF}=80^{\circ }}
979:{\displaystyle \angle {BCF}=50^{\circ }}
866:{\displaystyle \angle {BGC}=80^{\circ }}
822:{\displaystyle \angle {CBG}=20^{\circ }}
778:{\displaystyle \angle {BCG}=80^{\circ }}
29:
1683:
518:. Such calculations can establish that
1733:, John Wiley & Sons, p. 180,
7:
1875:SIAM Journal on Discrete Mathematics
1816:
1814:
1689:
1687:
1964:Introducing "3 circumcenter method"
549:is within any desired precision of
1588:
1533:
1353:
1309:
1265:
1150:
1037:
993:
949:
836:
792:
748:
525:
490:
433:
126:
109:
25:
483:The problem of calculating angle
1645:adventitious quadrangles problem
732:(See figure on the lower right.)
1:
1961:Saito, Hiroshi (2016-12-11),
1945:(in Japanese), archived from
514:is a standard application of
47:Langley's Adventitious Angles
34:Langley's Adventitious Angles
18:Langley’s Adventitious Angles
542:{\displaystyle \angle {BEF}}
507:{\displaystyle \angle {BEF}}
633:{\displaystyle 20^{\circ }}
569:{\displaystyle 30^{\circ }}
342:{\displaystyle 20^{\circ }}
216:{\displaystyle 30^{\circ }}
2018:
1887:10.1137/S0895480195281246
1674:Rigby in his 1978 paper.
1939:aerile_re (2015-10-27),
1824:The Mathematical Gazette
1757:The Mathematical Gazette
1700:The Mathematical Gazette
56:The Mathematical Gazette
1908:Saito, Hiroshi (2016),
1796:"The 80-80-20 Triangle"
1697:(1922), "Problem 644",
1651:adventitious quadrangle
1646:
1622:
1567:
1520:
1494:
1445:
1444:{\displaystyle GB=GE.}
1410:
1384:
1340:
1296:
1239:
1213:
1181:
1129:
1128:{\displaystyle BC=BF.}
1094:
1068:
1024:
980:
928:
927:{\displaystyle BC=BG.}
893:
867:
823:
779:
726:
700:
680:
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634:
607:
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543:
508:
467:
412:
389:
366:
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263:
240:
217:
190:
160:
92:
43:
35:
1644:
1623:
1568:
1521:
1495:
1493:{\displaystyle GE=GF}
1446:
1411:
1385:
1341:
1297:
1240:
1214:
1212:{\displaystyle BF=BG}
1182:
1130:
1095:
1069:
1025:
981:
929:
894:
868:
824:
780:
727:
701:
681:
658:
635:
608:
571:
544:
509:
468:
413:
390:
367:
344:
317:
287:
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241:
218:
191:
161:
93:
41:
33:
1800:www.cut-the-knot.org
1792:Bogomolny, Alexander
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522:
487:
430:
399:
376:
353:
326:
303:
273:
250:
227:
200:
177:
106:
76:
1914:GendaisĹ«gaku (現代数ĺ¦)
1670:and its diagonals.
1519:{\displaystyle EFG}
1409:{\displaystyle BGE}
1238:{\displaystyle BGF}
1093:{\displaystyle BCF}
892:{\displaystyle BCG}
725:{\displaystyle FG.}
91:{\displaystyle ABC}
51:Edward Mann Langley
1647:
1618:
1563:
1526:is isosceles with
1516:
1490:
1441:
1416:is isosceles with
1406:
1380:
1336:
1292:
1235:
1209:
1177:
1125:
1100:is isosceles with
1090:
1064:
1020:
976:
924:
899:is isosceles with
889:
863:
819:
775:
722:
696:
679:{\displaystyle AC}
676:
656:{\displaystyle BC}
653:
630:
606:{\displaystyle BG}
603:
566:
539:
516:Hansen's resection
504:
463:
411:{\displaystyle E.}
408:
388:{\displaystyle AC}
385:
365:{\displaystyle AB}
362:
339:
315:{\displaystyle BE}
312:
285:{\displaystyle F.}
282:
262:{\displaystyle AB}
259:
239:{\displaystyle AC}
236:
213:
189:{\displaystyle CF}
186:
156:
100:isosceles triangle
88:
44:
36:
2002:Triangle problems
699:{\displaystyle G}
16:(Redirected from
2009:
1973:
1967:
1958:
1952:
1950:
1936:
1930:
1928:
1905:
1899:
1897:
1872:
1863:
1857:
1855:
1831:(421): 183–191,
1818:
1809:
1808:
1807:
1806:
1788:
1782:
1780:
1751:
1745:
1743:
1721:
1710:
1708:
1691:
1655:rational numbers
1627:
1625:
1624:
1619:
1614:
1613:
1601:
1572:
1570:
1569:
1564:
1559:
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1517:
1499:
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1491:
1450:
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1442:
1415:
1413:
1412:
1407:
1389:
1387:
1386:
1381:
1379:
1378:
1366:
1345:
1343:
1342:
1337:
1335:
1334:
1322:
1301:
1299:
1298:
1293:
1291:
1290:
1278:
1244:
1242:
1241:
1236:
1218:
1216:
1215:
1210:
1186:
1184:
1183:
1178:
1176:
1175:
1163:
1134:
1132:
1131:
1126:
1099:
1097:
1096:
1091:
1073:
1071:
1070:
1065:
1063:
1062:
1050:
1029:
1027:
1026:
1021:
1019:
1018:
1006:
985:
983:
982:
977:
975:
974:
962:
933:
931:
930:
925:
898:
896:
895:
890:
872:
870:
869:
864:
862:
861:
849:
828:
826:
825:
820:
818:
817:
805:
784:
782:
781:
776:
774:
773:
761:
731:
729:
728:
723:
705:
703:
702:
697:
685:
683:
682:
677:
662:
660:
659:
654:
639:
637:
636:
631:
629:
628:
612:
610:
609:
604:
575:
573:
572:
567:
565:
564:
548:
546:
545:
540:
538:
513:
511:
510:
505:
503:
472:
470:
469:
464:
459:
458:
446:
417:
415:
414:
409:
394:
392:
391:
386:
371:
369:
368:
363:
348:
346:
345:
340:
338:
337:
321:
319:
318:
313:
291:
289:
288:
283:
268:
266:
265:
260:
245:
243:
242:
237:
222:
220:
219:
214:
212:
211:
195:
193:
192:
187:
165:
163:
162:
157:
152:
151:
139:
122:
97:
95:
94:
89:
21:
2017:
2016:
2012:
2011:
2010:
2008:
2007:
2006:
1992:
1991:
1982:
1977:
1976:
1960:
1959:
1955:
1938:
1937:
1933:
1916:(in Japanese),
1907:
1906:
1902:
1870:
1865:
1864:
1860:
1837:10.2307/3616687
1820:
1819:
1812:
1804:
1802:
1790:
1789:
1785:
1770:10.2307/3616644
1764:(408): 98–106,
1753:
1752:
1748:
1741:
1723:
1722:
1713:
1693:
1692:
1685:
1680:
1639:
1605:
1583:
1582:
1550:
1528:
1527:
1502:
1501:
1470:
1469:
1418:
1417:
1392:
1391:
1370:
1348:
1347:
1326:
1304:
1303:
1282:
1260:
1259:
1221:
1220:
1189:
1188:
1167:
1145:
1144:
1102:
1101:
1076:
1075:
1054:
1032:
1031:
1010:
988:
987:
966:
944:
943:
901:
900:
875:
874:
853:
831:
830:
809:
787:
786:
765:
743:
742:
708:
707:
688:
687:
665:
664:
642:
641:
620:
615:
614:
592:
591:
556:
551:
550:
520:
519:
485:
484:
481:
450:
428:
427:
397:
396:
374:
373:
351:
350:
329:
324:
323:
301:
300:
271:
270:
248:
247:
225:
224:
203:
198:
197:
175:
174:
143:
104:
103:
74:
73:
65:
28:
27:Geometry puzzle
23:
22:
15:
12:
11:
5:
2015:
2013:
2005:
2004:
1994:
1993:
1990:
1989:
1981:
1980:External links
1978:
1975:
1974:
1953:
1931:
1920:(590): 66–73,
1900:
1881:(1): 135–156,
1858:
1810:
1783:
1746:
1739:
1725:Darling, David
1711:
1695:Langley, E. M.
1682:
1681:
1679:
1676:
1638:
1637:Generalization
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1281:
1277:
1274:
1271:
1267:
1253:
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1251:
1250:
1234:
1231:
1228:
1219:then triangle
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1205:
1202:
1199:
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1166:
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672:
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627:
623:
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563:
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537:
534:
531:
527:
502:
499:
496:
492:
480:
477:
476:
475:
474:
473:
462:
457:
453:
449:
445:
442:
439:
435:
421:
420:
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418:
407:
404:
384:
381:
361:
358:
336:
332:
311:
308:
295:
294:
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281:
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235:
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210:
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87:
84:
81:
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26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2014:
2003:
2000:
1999:
1997:
1987:
1986:Angular Angst
1984:
1983:
1979:
1971:
1966:
1965:
1957:
1954:
1949:on 2016-04-16
1948:
1944:
1943:
1935:
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1911:
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1880:
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1834:
1830:
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1825:
1817:
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1811:
1801:
1797:
1793:
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1779:
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1740:9780471270478
1736:
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1457:
1456:
1455:
1438:
1435:
1432:
1429:
1426:
1423:
1403:
1400:
1397:
1390:and triangle
1375:
1371:
1367:
1363:
1360:
1357:
1331:
1327:
1323:
1319:
1316:
1313:
1287:
1283:
1279:
1275:
1272:
1269:
1257:
1256:
1255:
1254:
1248:
1232:
1229:
1226:
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1200:
1197:
1194:
1172:
1168:
1164:
1160:
1157:
1154:
1142:
1141:
1140:
1139:
1122:
1119:
1116:
1113:
1110:
1107:
1087:
1084:
1081:
1074:and triangle
1059:
1055:
1051:
1047:
1044:
1041:
1015:
1011:
1007:
1003:
1000:
997:
971:
967:
963:
959:
956:
953:
941:
940:
939:
938:
921:
918:
915:
912:
909:
906:
886:
883:
880:
873:and triangle
858:
854:
850:
846:
843:
840:
814:
810:
806:
802:
799:
796:
770:
766:
762:
758:
755:
752:
740:
739:
738:
737:
719:
716:
713:
693:
673:
670:
663:intersecting
650:
647:
625:
621:
600:
597:
589:
588:
587:
586:
585:
582:
577:
561:
557:
535:
532:
529:
517:
500:
497:
494:
478:
460:
455:
451:
447:
443:
440:
437:
425:
424:
423:
422:
405:
402:
382:
379:
359:
356:
334:
330:
309:
306:
299:
298:
297:
296:
279:
276:
256:
253:
233:
230:
208:
204:
183:
180:
173:
172:
171:
170:
153:
148:
144:
140:
136:
133:
130:
123:
119:
116:
113:
101:
85:
82:
79:
72:
71:
70:
69:
68:
62:
60:
58:
57:
52:
48:
40:
32:
19:
1970:Gendaisūgaku
1969:
1963:
1956:
1947:the original
1941:
1934:
1917:
1913:
1903:
1878:
1874:
1861:
1828:
1822:
1803:, retrieved
1799:
1786:
1761:
1755:
1749:
1729:
1704:
1698:
1672:
1668:triacontagon
1660:
1650:
1648:
1632:
581:James Mercer
578:
482:
66:
54:
46:
45:
1988:, MathPages
1247:equilateral
63:The problem
1805:2018-06-03
1678:References
1663:Gerrit Bol
1581:Therefore
1926:2187-6495
1611:∘
1589:∠
1556:∘
1534:∠
1500:triangle
1376:∘
1354:∠
1332:∘
1310:∠
1288:∘
1266:∠
1173:∘
1151:∠
1060:∘
1038:∠
1016:∘
994:∠
972:∘
950:∠
859:∘
837:∠
815:∘
793:∠
771:∘
749:∠
706:and draw
626:∘
562:∘
526:∠
491:∠
456:∘
434:∠
335:∘
209:∘
149:∘
127:∠
110:∠
59:in 1922.
1996:Category
1727:(2004),
479:Solution
1972:(現代数ĺ¦).
1895:8673508
1853:0513855
1845:3616687
1778:3616644
1924:
1893:
1851:
1843:
1776:
1737:
1468:Since
1258:Since
1143:Since
942:Since
741:Since
426:Prove
98:is an
1891:S2CID
1871:(PDF)
1841:JSTOR
1774:JSTOR
1707:: 173
1346:then
1030:then
829:then
590:Draw
372:cuts
246:cuts
102:with
1922:ISSN
1735:ISBN
1302:and
1187:and
986:and
785:and
1883:doi
1833:doi
1766:doi
1284:100
1245:is
686:at
640:to
613:at
395:in
349:to
322:at
269:in
223:to
196:at
53:in
1998::
1918:49
1912:,
1889:,
1879:11
1877:,
1873:,
1849:MR
1847:,
1839:,
1829:62
1827:,
1813:^
1798:,
1794:,
1772:,
1762:59
1760:,
1714:^
1705:11
1703:,
1686:^
1607:30
1552:70
1372:40
1328:40
1169:60
1056:50
1012:80
968:50
855:80
811:20
767:80
622:20
558:30
452:30
331:20
205:30
145:80
1951:.
1929:.
1898:.
1885::
1856:.
1835::
1781:.
1768::
1744:.
1709:.
1616:.
1603:=
1599:F
1596:E
1593:B
1561:.
1548:=
1544:F
1541:E
1538:G
1514:G
1511:F
1508:E
1488:F
1485:G
1482:=
1479:E
1476:G
1439:.
1436:E
1433:G
1430:=
1427:B
1424:G
1404:E
1401:G
1398:B
1368:=
1364:B
1361:E
1358:G
1324:=
1320:E
1317:B
1314:G
1280:=
1276:E
1273:G
1270:B
1249:.
1233:F
1230:G
1227:B
1207:G
1204:B
1201:=
1198:F
1195:B
1165:=
1161:G
1158:B
1155:F
1123:.
1120:F
1117:B
1114:=
1111:C
1108:B
1088:F
1085:C
1082:B
1052:=
1048:C
1045:F
1042:B
1008:=
1004:F
1001:B
998:C
964:=
960:F
957:C
954:B
922:.
919:G
916:B
913:=
910:C
907:B
887:G
884:C
881:B
851:=
847:C
844:G
841:B
807:=
803:G
800:B
797:C
763:=
759:G
756:C
753:B
720:.
717:G
714:F
694:G
674:C
671:A
651:C
648:B
601:G
598:B
536:F
533:E
530:B
501:F
498:E
495:B
461:.
448:=
444:F
441:E
438:B
406:.
403:E
383:C
380:A
360:B
357:A
310:E
307:B
280:.
277:F
257:B
254:A
234:C
231:A
184:F
181:C
154:.
141:=
137:B
134:C
131:A
124:=
120:A
117:B
114:C
86:C
83:B
80:A
20:)
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