28:
1631:
20:
1662:
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
1654:
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
1646:
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.
572:
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.
153:
1650:
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
1615:
1560:
1289:
460:
1377:
1333:
1174:
1061:
1017:
973:
860:
816:
772:
1622:
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.
536:
501:
627:
563:
336:
210:
1438:
1122:
921:
1487:
1206:
1513:
1403:
1232:
1087:
886:
719:
85:
673:
650:
600:
405:
382:
359:
309:
279:
256:
233:
183:
693:
1727:
1929:
94:
1655:
Poonen and
Michael Rubinstein in 1998. The article contains a history of the problem and a picture featuring the regular
569:
1990:
1812:
1745:
1688:
1573:
1518:
1250:
418:
44:
1338:
1294:
1135:
1022:
978:
934:
821:
777:
733:
38:
is a puzzle in which one must infer an angle in a geometric diagram from other given angles. It was posed by
1780:
1683:
504:
39:
510:
475:
1879:
1829:
1762:
605:
541:
314:
188:
88:
1951:
27:
1910:
1723:
1717:
1784:
1871:
1821:
1754:
1713:
1408:
1092:
891:
1841:
1630:
1460:
1179:
1837:
1643:
1642:
when the angles between its diagonals and sides are all rational angles, angles that give
1492:
1382:
1211:
1066:
865:
698:
64:
655:
632:
582:
387:
364:
341:
291:
261:
238:
215:
165:
678:
1974:
1898:
1984:
1935:
1883:
1656:
19:
1856:
1235:
565:, but being of only finite precision, always leave doubt about the exact value.
1899:"The adventitious quadrangles was solved completely by the elementary solution"
1875:
1857:"The number of intersection points made by the diagonals of a regular polygon"
1651:
1914:
1810:
Rigby, J. F. (1978), "Adventitious quadrangles: a geometrical approach",
1833:
1766:
1719:
The
Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes
1825:
1758:
1629:
26:
568:
A direct proof using classical geometry was developed by
148:{\displaystyle \angle {CBA}=\angle {ACB}=80^{\circ }.}
1576:
1521:
1495:
1463:
1411:
1385:
1341:
1297:
1253:
1214:
1182:
1138:
1095:
1069:
1025:
981:
937:
894:
868:
824:
780:
736:
701:
681:
658:
635:
608:
585:
544:
513:
478:
421:
390:
367:
344:
317:
294:
264:
241:
218:
191:
168:
97:
67:
1448:
Therefore all the red lines in the figure are equal.
1609:
1554:
1507:
1481:
1432:
1397:
1371:
1327:
1283:
1226:
1200:
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915:
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810:
766:
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687:
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621:
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353:
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273:
250:
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204:
177:
147:
79:
56:In its original form the problem was as follows:
31:Solution to Langley's 80-80-20 triangle problem
8:
1743:Tripp, Colin (1975), "Adventitious angles",
1855:Poonen, Bjorn; Rubinstein, Michael (1998),
1957:- English translation of the article from
1931:The last challenge from Geometry the Great
1638:A quadrilateral such as BCEF is called an
1708:
1706:
1704:
1610:{\displaystyle \angle {BEF}=30^{\circ }.}
1598:
1580:
1575:
1555:{\displaystyle \angle {GEF}=70^{\circ }.}
1543:
1525:
1520:
1494:
1462:
1410:
1384:
1363:
1345:
1340:
1319:
1301:
1296:
1284:{\displaystyle \angle {BGE}=100^{\circ }}
1275:
1257:
1252:
1213:
1181:
1160:
1142:
1137:
1094:
1068:
1047:
1029:
1024:
1003:
985:
980:
959:
941:
936:
893:
867:
846:
828:
823:
802:
784:
779:
758:
740:
735:
700:
680:
657:
634:
613:
607:
584:
549:
543:
517:
512:
482:
477:
455:{\displaystyle \angle {BEF}=30^{\circ }.}
443:
425:
420:
389:
366:
343:
322:
316:
293:
263:
240:
217:
196:
190:
167:
136:
118:
101:
96:
66:
1372:{\displaystyle \angle {GEB}=40^{\circ }}
1328:{\displaystyle \angle {GBE}=40^{\circ }}
1169:{\displaystyle \angle {FBG}=60^{\circ }}
1056:{\displaystyle \angle {BFC}=50^{\circ }}
1012:{\displaystyle \angle {CBF}=80^{\circ }}
968:{\displaystyle \angle {BCF}=50^{\circ }}
855:{\displaystyle \angle {BGC}=80^{\circ }}
811:{\displaystyle \angle {CBG}=20^{\circ }}
767:{\displaystyle \angle {BCG}=80^{\circ }}
18:
1672:
507:. Such calculations can establish that
1722:, John Wiley & Sons, p. 180,
7:
1864:SIAM Journal on Discrete Mathematics
1805:
1803:
1678:
1676:
1953:Introducing "3 circumcenter method"
538:is within any desired precision of
1577:
1522:
1342:
1298:
1254:
1139:
1026:
982:
938:
825:
781:
737:
514:
479:
422:
115:
98:
14:
472:The problem of calculating angle
1634:adventitious quadrangles problem
721:(See figure on the lower right.)
1:
1950:Saito, Hiroshi (2016-12-11),
1934:(in Japanese), archived from
503:is a standard application of
36:Langley's Adventitious Angles
23:Langley's Adventitious Angles
531:{\displaystyle \angle {BEF}}
496:{\displaystyle \angle {BEF}}
622:{\displaystyle 20^{\circ }}
558:{\displaystyle 30^{\circ }}
331:{\displaystyle 20^{\circ }}
205:{\displaystyle 30^{\circ }}
2007:
1876:10.1137/S0895480195281246
1663:Rigby in his 1978 paper.
1928:aerile_re (2015-10-27),
1813:The Mathematical Gazette
1746:The Mathematical Gazette
1689:The Mathematical Gazette
45:The Mathematical Gazette
1897:Saito, Hiroshi (2016),
1785:"The 80-80-20 Triangle"
1686:(1922), "Problem 644",
1640:adventitious quadrangle
1635:
1611:
1556:
1509:
1483:
1434:
1433:{\displaystyle GB=GE.}
1399:
1373:
1329:
1285:
1228:
1202:
1170:
1118:
1117:{\displaystyle BC=BF.}
1083:
1057:
1013:
969:
917:
916:{\displaystyle BC=BG.}
882:
856:
812:
768:
715:
689:
669:
646:
623:
596:
559:
532:
497:
456:
401:
378:
355:
332:
305:
275:
252:
229:
206:
179:
149:
81:
32:
24:
1633:
1612:
1557:
1510:
1484:
1482:{\displaystyle GE=GF}
1435:
1400:
1374:
1330:
1286:
1229:
1203:
1201:{\displaystyle BF=BG}
1171:
1119:
1084:
1058:
1014:
970:
918:
883:
857:
813:
769:
716:
690:
670:
647:
624:
597:
560:
533:
498:
457:
402:
379:
356:
333:
306:
276:
253:
230:
207:
180:
150:
82:
30:
22:
1789:www.cut-the-knot.org
1781:Bogomolny, Alexander
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606:
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542:
511:
476:
419:
388:
365:
342:
315:
292:
262:
239:
216:
189:
166:
95:
65:
1903:Gendaisūgaku (現代数学)
1659:and its diagonals.
1508:{\displaystyle EFG}
1398:{\displaystyle BGE}
1227:{\displaystyle BGF}
1082:{\displaystyle BCF}
881:{\displaystyle BCG}
714:{\displaystyle FG.}
80:{\displaystyle ABC}
40:Edward Mann Langley
1636:
1607:
1552:
1515:is isosceles with
1505:
1479:
1430:
1405:is isosceles with
1395:
1369:
1325:
1281:
1224:
1198:
1166:
1114:
1089:is isosceles with
1079:
1053:
1009:
965:
913:
888:is isosceles with
878:
852:
808:
764:
711:
685:
668:{\displaystyle AC}
665:
645:{\displaystyle BC}
642:
619:
595:{\displaystyle BG}
592:
555:
528:
505:Hansen's resection
493:
452:
400:{\displaystyle E.}
397:
377:{\displaystyle AC}
374:
354:{\displaystyle AB}
351:
328:
304:{\displaystyle BE}
301:
274:{\displaystyle F.}
271:
251:{\displaystyle AB}
248:
228:{\displaystyle AC}
225:
202:
178:{\displaystyle CF}
175:
145:
89:isosceles triangle
77:
33:
25:
1991:Triangle problems
688:{\displaystyle G}
1998:
1962:
1956:
1947:
1941:
1939:
1925:
1919:
1917:
1894:
1888:
1886:
1861:
1852:
1846:
1844:
1820:(421): 183–191,
1807:
1798:
1797:
1796:
1795:
1777:
1771:
1769:
1740:
1734:
1732:
1710:
1699:
1697:
1680:
1644:rational numbers
1616:
1614:
1613:
1608:
1603:
1602:
1590:
1561:
1559:
1558:
1553:
1548:
1547:
1535:
1514:
1512:
1511:
1506:
1488:
1486:
1485:
1480:
1439:
1437:
1436:
1431:
1404:
1402:
1401:
1396:
1378:
1376:
1375:
1370:
1368:
1367:
1355:
1334:
1332:
1331:
1326:
1324:
1323:
1311:
1290:
1288:
1287:
1282:
1280:
1279:
1267:
1233:
1231:
1230:
1225:
1207:
1205:
1204:
1199:
1175:
1173:
1172:
1167:
1165:
1164:
1152:
1123:
1121:
1120:
1115:
1088:
1086:
1085:
1080:
1062:
1060:
1059:
1054:
1052:
1051:
1039:
1018:
1016:
1015:
1010:
1008:
1007:
995:
974:
972:
971:
966:
964:
963:
951:
922:
920:
919:
914:
887:
885:
884:
879:
861:
859:
858:
853:
851:
850:
838:
817:
815:
814:
809:
807:
806:
794:
773:
771:
770:
765:
763:
762:
750:
720:
718:
717:
712:
694:
692:
691:
686:
674:
672:
671:
666:
651:
649:
648:
643:
628:
626:
625:
620:
618:
617:
601:
599:
598:
593:
564:
562:
561:
556:
554:
553:
537:
535:
534:
529:
527:
502:
500:
499:
494:
492:
461:
459:
458:
453:
448:
447:
435:
406:
404:
403:
398:
383:
381:
380:
375:
360:
358:
357:
352:
337:
335:
334:
329:
327:
326:
310:
308:
307:
302:
280:
278:
277:
272:
257:
255:
254:
249:
234:
232:
231:
226:
211:
209:
208:
203:
201:
200:
184:
182:
181:
176:
154:
152:
151:
146:
141:
140:
128:
111:
86:
84:
83:
78:
2006:
2005:
2001:
2000:
1999:
1997:
1996:
1995:
1981:
1980:
1971:
1966:
1965:
1949:
1948:
1944:
1927:
1926:
1922:
1905:(in Japanese),
1896:
1895:
1891:
1859:
1854:
1853:
1849:
1826:10.2307/3616687
1809:
1808:
1801:
1793:
1791:
1779:
1778:
1774:
1759:10.2307/3616644
1753:(408): 98–106,
1742:
1741:
1737:
1730:
1712:
1711:
1702:
1682:
1681:
1674:
1669:
1628:
1594:
1572:
1571:
1539:
1517:
1516:
1491:
1490:
1459:
1458:
1407:
1406:
1381:
1380:
1359:
1337:
1336:
1315:
1293:
1292:
1271:
1249:
1248:
1210:
1209:
1178:
1177:
1156:
1134:
1133:
1091:
1090:
1065:
1064:
1043:
1021:
1020:
999:
977:
976:
955:
933:
932:
890:
889:
864:
863:
842:
820:
819:
798:
776:
775:
754:
732:
731:
697:
696:
677:
676:
654:
653:
631:
630:
609:
604:
603:
581:
580:
545:
540:
539:
509:
508:
474:
473:
470:
439:
417:
416:
386:
385:
363:
362:
340:
339:
318:
313:
312:
290:
289:
260:
259:
237:
236:
214:
213:
192:
187:
186:
164:
163:
132:
93:
92:
63:
62:
54:
17:
16:Geometry puzzle
12:
11:
5:
2004:
2002:
1994:
1993:
1983:
1982:
1979:
1978:
1970:
1969:External links
1967:
1964:
1963:
1942:
1920:
1909:(590): 66–73,
1889:
1870:(1): 135–156,
1847:
1799:
1772:
1735:
1728:
1714:Darling, David
1700:
1684:Langley, E. M.
1671:
1670:
1668:
1665:
1627:
1626:Generalization
1624:
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1362:
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1314:
1310:
1307:
1304:
1300:
1278:
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1270:
1266:
1263:
1260:
1256:
1242:
1241:
1240:
1239:
1223:
1220:
1217:
1208:then triangle
1197:
1194:
1191:
1188:
1185:
1163:
1159:
1155:
1151:
1148:
1145:
1141:
1127:
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1125:
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1110:
1107:
1104:
1101:
1098:
1078:
1075:
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1038:
1035:
1032:
1028:
1006:
1002:
998:
994:
991:
988:
984:
962:
958:
954:
950:
947:
944:
940:
926:
925:
924:
923:
912:
909:
906:
903:
900:
897:
877:
874:
871:
849:
845:
841:
837:
834:
831:
827:
805:
801:
797:
793:
790:
787:
783:
761:
757:
753:
749:
746:
743:
739:
725:
724:
723:
722:
710:
707:
704:
684:
664:
661:
641:
638:
616:
612:
591:
588:
552:
548:
526:
523:
520:
516:
491:
488:
485:
481:
469:
466:
465:
464:
463:
462:
451:
446:
442:
438:
434:
431:
428:
424:
410:
409:
408:
407:
396:
393:
373:
370:
350:
347:
325:
321:
300:
297:
284:
283:
282:
281:
270:
267:
247:
244:
224:
221:
199:
195:
174:
171:
158:
157:
156:
155:
144:
139:
135:
131:
127:
124:
121:
117:
114:
110:
107:
104:
100:
76:
73:
70:
53:
50:
15:
13:
10:
9:
6:
4:
3:
2:
2003:
1992:
1989:
1988:
1986:
1976:
1975:Angular Angst
1973:
1972:
1968:
1960:
1955:
1954:
1946:
1943:
1938:on 2016-04-16
1937:
1933:
1932:
1924:
1921:
1916:
1912:
1908:
1904:
1900:
1893:
1890:
1885:
1881:
1877:
1873:
1869:
1865:
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1851:
1848:
1843:
1839:
1835:
1831:
1827:
1823:
1819:
1815:
1814:
1806:
1804:
1800:
1790:
1786:
1782:
1776:
1773:
1768:
1764:
1760:
1756:
1752:
1748:
1747:
1739:
1736:
1731:
1729:9780471270478
1725:
1721:
1720:
1715:
1709:
1707:
1705:
1701:
1695:
1691:
1690:
1685:
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1641:
1632:
1625:
1623:
1604:
1599:
1595:
1591:
1587:
1584:
1581:
1569:
1568:
1567:
1566:
1549:
1544:
1540:
1536:
1532:
1529:
1526:
1502:
1499:
1496:
1476:
1473:
1470:
1467:
1464:
1456:
1455:
1454:
1453:
1447:
1446:
1445:
1444:
1427:
1424:
1421:
1418:
1415:
1412:
1392:
1389:
1386:
1379:and triangle
1364:
1360:
1356:
1352:
1349:
1346:
1320:
1316:
1312:
1308:
1305:
1302:
1276:
1272:
1268:
1264:
1261:
1258:
1246:
1245:
1244:
1243:
1237:
1221:
1218:
1215:
1195:
1192:
1189:
1186:
1183:
1161:
1157:
1153:
1149:
1146:
1143:
1131:
1130:
1129:
1128:
1111:
1108:
1105:
1102:
1099:
1096:
1076:
1073:
1070:
1063:and triangle
1048:
1044:
1040:
1036:
1033:
1030:
1004:
1000:
996:
992:
989:
986:
960:
956:
952:
948:
945:
942:
930:
929:
928:
927:
910:
907:
904:
901:
898:
895:
875:
872:
869:
862:and triangle
847:
843:
839:
835:
832:
829:
803:
799:
795:
791:
788:
785:
759:
755:
751:
747:
744:
741:
729:
728:
727:
726:
708:
705:
702:
682:
662:
659:
652:intersecting
639:
636:
614:
610:
589:
586:
578:
577:
576:
575:
574:
571:
566:
550:
546:
524:
521:
518:
506:
489:
486:
483:
467:
449:
444:
440:
436:
432:
429:
426:
414:
413:
412:
411:
394:
391:
371:
368:
348:
345:
323:
319:
298:
295:
288:
287:
286:
285:
268:
265:
245:
242:
222:
219:
197:
193:
172:
169:
162:
161:
160:
159:
142:
137:
133:
129:
125:
122:
119:
112:
108:
105:
102:
90:
74:
71:
68:
61:
60:
59:
58:
57:
51:
49:
47:
46:
41:
37:
29:
21:
1959:Gendaisūgaku
1958:
1952:
1945:
1936:the original
1930:
1923:
1906:
1902:
1892:
1867:
1863:
1850:
1817:
1811:
1792:, retrieved
1788:
1775:
1750:
1744:
1738:
1718:
1693:
1687:
1661:
1657:triacontagon
1649:
1639:
1637:
1621:
570:James Mercer
567:
471:
55:
43:
35:
34:
1977:, MathPages
1236:equilateral
52:The problem
1794:2018-06-03
1667:References
1652:Gerrit Bol
1570:Therefore
1915:2187-6495
1600:∘
1578:∠
1545:∘
1523:∠
1489:triangle
1365:∘
1343:∠
1321:∘
1299:∠
1277:∘
1255:∠
1162:∘
1140:∠
1049:∘
1027:∠
1005:∘
983:∠
961:∘
939:∠
848:∘
826:∠
804:∘
782:∠
760:∘
738:∠
695:and draw
615:∘
551:∘
515:∠
480:∠
445:∘
423:∠
324:∘
198:∘
138:∘
116:∠
99:∠
48:in 1922.
1985:Category
1716:(2004),
468:Solution
1961:(現代数学).
1884:8673508
1842:0513855
1834:3616687
1767:3616644
1913:
1882:
1840:
1832:
1765:
1726:
1457:Since
1247:Since
1132:Since
931:Since
730:Since
415:Prove
87:is an
1880:S2CID
1860:(PDF)
1830:JSTOR
1763:JSTOR
1696:: 173
1335:then
1019:then
818:then
579:Draw
361:cuts
235:cuts
91:with
1911:ISSN
1724:ISBN
1291:and
1176:and
975:and
774:and
1872:doi
1822:doi
1755:doi
1273:100
1234:is
675:at
629:to
602:at
384:in
338:to
311:at
258:in
212:to
185:at
42:in
1987::
1907:49
1901:,
1878:,
1868:11
1866:,
1862:,
1838:MR
1836:,
1828:,
1818:62
1816:,
1802:^
1787:,
1783:,
1761:,
1751:59
1749:,
1703:^
1694:11
1692:,
1675:^
1596:30
1541:70
1361:40
1317:40
1158:60
1045:50
1001:80
957:50
844:80
800:20
756:80
611:20
547:30
441:30
320:20
194:30
134:80
1940:.
1918:.
1887:.
1874::
1845:.
1824::
1770:.
1757::
1733:.
1698:.
1605:.
1592:=
1588:F
1585:E
1582:B
1550:.
1537:=
1533:F
1530:E
1527:G
1503:G
1500:F
1497:E
1477:F
1474:G
1471:=
1468:E
1465:G
1428:.
1425:E
1422:G
1419:=
1416:B
1413:G
1393:E
1390:G
1387:B
1357:=
1353:B
1350:E
1347:G
1313:=
1309:E
1306:B
1303:G
1269:=
1265:E
1262:G
1259:B
1238:.
1222:F
1219:G
1216:B
1196:G
1193:B
1190:=
1187:F
1184:B
1154:=
1150:G
1147:B
1144:F
1112:.
1109:F
1106:B
1103:=
1100:C
1097:B
1077:F
1074:C
1071:B
1041:=
1037:C
1034:F
1031:B
997:=
993:F
990:B
987:C
953:=
949:F
946:C
943:B
911:.
908:G
905:B
902:=
899:C
896:B
876:G
873:C
870:B
840:=
836:C
833:G
830:B
796:=
792:G
789:B
786:C
752:=
748:G
745:C
742:B
709:.
706:G
703:F
683:G
663:C
660:A
640:C
637:B
590:G
587:B
525:F
522:E
519:B
490:F
487:E
484:B
450:.
437:=
433:F
430:E
427:B
395:.
392:E
372:C
369:A
349:B
346:A
299:E
296:B
269:.
266:F
246:B
243:A
223:C
220:A
173:F
170:C
143:.
130:=
126:B
123:C
120:A
113:=
109:A
106:B
103:C
75:C
72:B
69:A
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