1382:
20:
74:
667:
630:
593:
1133:
1103:
1073:
1303:
1276:
1249:
974:
947:
920:
556:
529:
502:
475:
379:
352:
325:
298:
251:
224:
1202:
1179:
1156:
1043:
1020:
997:
448:
425:
402:
1346:
1326:
1222:
893:
873:
853:
833:
813:
793:
773:
753:
733:
713:
271:
193:
173:
153:
133:
113:
1610:
1657:
1647:
1587:
1553:
1420:
through P such that they meet the conic at A, A'; B, B' ; C, C' respectively. Let D be a point on the
1585:, Global Journal of Advanced Research on Classical and Modern Geometries, Vol.4, (2015), Issue 2, page 102-105
1652:
558:
intersects those side lines. The Droz-Farny line theorem says that the midpoints of the three segments
26:
689:
1357:
196:
1523:
1626:
81:
1599:
1569:
1536:
René Goormaghtigh (1930), "Sur une généralisation du théoreme de Noyer, Droz-Farny et
Neuberg".
1492:
635:
598:
561:
1370:
be a point on the plane, let three parallel segments AA', BB', CC' such that its midpoints and
1595:
1565:
1488:
677:
1480:
1401:
1397:
1108:
1078:
1048:
1281:
1254:
1227:
952:
925:
898:
534:
507:
480:
453:
357:
330:
303:
276:
229:
202:
1591:
1557:
1405:
1184:
1161:
1138:
1025:
1002:
979:
430:
407:
384:
1527:. The MacTutor History of Mathematics archive. Online document, accessed on 2014-10-05.
1421:
1331:
1311:
1207:
878:
858:
838:
818:
798:
778:
758:
738:
718:
698:
256:
178:
158:
138:
118:
98:
1505:
1641:
1630:
1381:
89:
19:
1510:
670:
1580:
1550:
1424:
of point P with respect to (S) or D lies on the conic (S). Let DA' ∩ BC =A
1562:
Global
Journal of Advanced Research on Classical and Modern Geometries
688:
A generalization of the Droz-Farny line theorem was proved in 1930 by
1615:. The Mathematical Gazette, 99, pp 339-341. doi:10.1017/mag.2015.47
1551:
A synthetic proof of Dao's generalization of
Goormaghtigh's theorem
1308:
The Droz-Farny line theorem is a special case of this result, when
1393:
1380:
18:
1481:
A Purely
Synthetic Proof of the Droz-Farny Line Theorem
1334:
1314:
1284:
1257:
1230:
1210:
1187:
1164:
1141:
1111:
1081:
1051:
1028:
1005:
982:
955:
928:
901:
881:
861:
841:
821:
801:
781:
761:
741:
721:
701:
680:
in 1899, but it is not clear whether he had a proof.
638:
601:
564:
537:
510:
483:
456:
433:
410:
387:
360:
333:
306:
279:
259:
232:
205:
181:
161:
141:
121:
101:
88:
is a property of two perpendicular lines through the
29:
16:
Property of perpendicular lines through orthocenters
1224:. Goormaghtigh's theorem then says that the points
1340:
1320:
1297:
1270:
1243:
1216:
1196:
1173:
1150:
1127:
1097:
1067:
1037:
1014:
991:
968:
941:
914:
887:
867:
847:
827:
807:
787:
767:
747:
727:
707:
661:
624:
587:
550:
523:
496:
469:
442:
419:
396:
373:
346:
319:
292:
265:
245:
218:
195:be its orthocenter (the common point of its three
187:
167:
147:
127:
107:
68:
253:be any two mutually perpendicular lines through
1204:, respectively, by reflection against the line
8:
1521:J. J. O'Connor and E. F. Robertson (2006),
1503:Floor van Lamoen and Eric W. Weisstein (),
1462:
1460:
1333:
1313:
1289:
1283:
1262:
1256:
1235:
1229:
1209:
1186:
1163:
1140:
1119:
1110:
1089:
1080:
1059:
1050:
1027:
1004:
981:
960:
954:
933:
927:
906:
900:
880:
860:
840:
820:
800:
780:
760:
740:
720:
700:
653:
643:
637:
616:
606:
600:
579:
569:
563:
542:
536:
515:
509:
488:
482:
461:
455:
432:
409:
386:
365:
359:
338:
332:
311:
305:
284:
278:
258:
237:
231:
210:
204:
180:
160:
140:
120:
100:
60:
47:
34:
28:
1466:A. Droz-Farny (1899), "Question 14111".
1378:respectively at three collinear points.
1456:
1374:are collinear. Then PA', PB', PC' meet
1356:The theorem was further generalized by
7:
1045:, respectively, such that the lines
450:, respectively. Similarly, let Let
14:
1360:. The generalization as follows:
69:{\displaystyle A_{0},B_{0},C_{0}}
1328:is the orthocenter of triangle
1613:99.20 A projective Simson line
1351:
683:
1:
684:Goormaghtigh's generalization
1487:, volume 14, pages 219–224,
1135:are the images of the lines
976:be points on the side lines
715:be a triangle with vertices
115:be a triangle with vertices
1564:, volume 3, pages 125–129,
1385:Dao's second generalization
795:be any point distinct from
1674:
1627:Two Pascals merge into one
676:The theorem was stated by
662:{\displaystyle C_{1}C_{2}}
625:{\displaystyle B_{1}B_{2}}
588:{\displaystyle A_{1}A_{2}}
381:intersects the side lines
92:of an arbitrary triangle.
1479:Jean-Louis Ayme (2004), "
1658:Theorems about triangles
1549:Son Tran Hoang (2014), "
1470:, volume 71, pages 89-90
1366:Let ABC be a triangle,
86:Droz-Farny line theorem
1625:O.T.Dao 29-July-2013,
1583:A proof of Dao theorem
1390:Second generalization:
1386:
1342:
1322:
1299:
1272:
1245:
1218:
1198:
1175:
1152:
1129:
1128:{\displaystyle PC_{1}}
1099:
1098:{\displaystyle PB_{1}}
1069:
1068:{\displaystyle PA_{1}}
1039:
1016:
993:
970:
943:
916:
889:
869:
849:
829:
809:
789:
769:
749:
729:
709:
663:
626:
589:
552:
525:
498:
471:
444:
421:
398:
375:
348:
321:
294:
267:
247:
220:
189:
169:
149:
129:
109:
77:
70:
1468:The Educational Times
1384:
1364:First generalization:
1343:
1323:
1300:
1298:{\displaystyle C_{1}}
1273:
1271:{\displaystyle B_{1}}
1246:
1244:{\displaystyle A_{1}}
1219:
1199:
1176:
1153:
1130:
1100:
1070:
1040:
1017:
994:
971:
969:{\displaystyle C_{1}}
944:
942:{\displaystyle B_{1}}
917:
915:{\displaystyle A_{1}}
890:
870:
850:
830:
810:
790:
770:
750:
730:
710:
664:
627:
590:
553:
551:{\displaystyle L_{2}}
526:
524:{\displaystyle C_{2}}
499:
497:{\displaystyle B_{2}}
472:
470:{\displaystyle A_{2}}
445:
422:
399:
376:
374:{\displaystyle L_{1}}
349:
347:{\displaystyle C_{1}}
322:
320:{\displaystyle B_{1}}
295:
293:{\displaystyle A_{1}}
268:
248:
246:{\displaystyle L_{2}}
221:
219:{\displaystyle L_{1}}
190:
170:
150:
130:
110:
71:
22:
1611:Geoff Smith (2015).
1540:, volume 44, page 25
1352:Dao's generalization
1332:
1312:
1282:
1255:
1228:
1208:
1185:
1162:
1139:
1109:
1079:
1049:
1026:
1003:
980:
953:
926:
899:
879:
875:be any line through
859:
839:
819:
799:
779:
759:
739:
719:
699:
636:
599:
562:
535:
531:be the points where
508:
481:
454:
431:
408:
385:
358:
354:be the points where
331:
304:
277:
257:
230:
203:
179:
159:
139:
119:
99:
27:
1581:Nguyen Ngoc Giang,
1485:Forum Geometricorum
1648:Euclidean geometry
1590:2014-10-06 at the
1556:2014-10-06 at the
1506:Droz-Farny Theorem
1404:. Construct three
1387:
1338:
1318:
1295:
1268:
1241:
1214:
1197:{\displaystyle PC}
1194:
1174:{\displaystyle PB}
1171:
1151:{\displaystyle PA}
1148:
1125:
1095:
1065:
1038:{\displaystyle AB}
1035:
1015:{\displaystyle CA}
1012:
992:{\displaystyle BC}
989:
966:
939:
912:
885:
865:
845:
825:
805:
785:
765:
745:
725:
705:
659:
622:
585:
548:
521:
494:
467:
443:{\displaystyle AB}
440:
420:{\displaystyle CA}
417:
397:{\displaystyle BC}
394:
371:
344:
317:
290:
263:
243:
216:
185:
165:
145:
125:
105:
82:Euclidean geometry
78:
76:is Droz-Farny line
66:
1524:Arnold Droz-Farny
1341:{\displaystyle T}
1321:{\displaystyle P}
1217:{\displaystyle L}
888:{\displaystyle P}
868:{\displaystyle L}
848:{\displaystyle C}
828:{\displaystyle B}
808:{\displaystyle A}
788:{\displaystyle P}
768:{\displaystyle C}
748:{\displaystyle B}
728:{\displaystyle A}
708:{\displaystyle T}
690:René Goormaghtigh
678:Arnold Droz-Farny
266:{\displaystyle H}
188:{\displaystyle H}
168:{\displaystyle C}
148:{\displaystyle B}
128:{\displaystyle A}
108:{\displaystyle T}
23:The line through
1665:
1633:
1623:
1617:
1608:
1602:
1578:
1572:
1547:
1541:
1534:
1528:
1519:
1513:
1501:
1495:
1477:
1471:
1464:
1448:are collinear.
1347:
1345:
1344:
1339:
1327:
1325:
1324:
1319:
1304:
1302:
1301:
1296:
1294:
1293:
1277:
1275:
1274:
1269:
1267:
1266:
1250:
1248:
1247:
1242:
1240:
1239:
1223:
1221:
1220:
1215:
1203:
1201:
1200:
1195:
1180:
1178:
1177:
1172:
1157:
1155:
1154:
1149:
1134:
1132:
1131:
1126:
1124:
1123:
1104:
1102:
1101:
1096:
1094:
1093:
1074:
1072:
1071:
1066:
1064:
1063:
1044:
1042:
1041:
1036:
1021:
1019:
1018:
1013:
998:
996:
995:
990:
975:
973:
972:
967:
965:
964:
948:
946:
945:
940:
938:
937:
921:
919:
918:
913:
911:
910:
894:
892:
891:
886:
874:
872:
871:
866:
854:
852:
851:
846:
834:
832:
831:
826:
814:
812:
811:
806:
794:
792:
791:
786:
774:
772:
771:
766:
754:
752:
751:
746:
734:
732:
731:
726:
714:
712:
711:
706:
668:
666:
665:
660:
658:
657:
648:
647:
631:
629:
628:
623:
621:
620:
611:
610:
594:
592:
591:
586:
584:
583:
574:
573:
557:
555:
554:
549:
547:
546:
530:
528:
527:
522:
520:
519:
503:
501:
500:
495:
493:
492:
476:
474:
473:
468:
466:
465:
449:
447:
446:
441:
426:
424:
423:
418:
403:
401:
400:
395:
380:
378:
377:
372:
370:
369:
353:
351:
350:
345:
343:
342:
326:
324:
323:
318:
316:
315:
299:
297:
296:
291:
289:
288:
272:
270:
269:
264:
252:
250:
249:
244:
242:
241:
225:
223:
222:
217:
215:
214:
194:
192:
191:
186:
174:
172:
171:
166:
154:
152:
151:
146:
134:
132:
131:
126:
114:
112:
111:
106:
75:
73:
72:
67:
65:
64:
52:
51:
39:
38:
1673:
1672:
1668:
1667:
1666:
1664:
1663:
1662:
1638:
1637:
1636:
1624:
1620:
1609:
1605:
1592:Wayback Machine
1579:
1575:
1558:Wayback Machine
1548:
1544:
1535:
1531:
1520:
1516:
1502:
1498:
1478:
1474:
1465:
1458:
1454:
1447:
1443:
1439:
1435:
1431:
1427:
1419:
1415:
1411:
1354:
1330:
1329:
1310:
1309:
1305:are collinear.
1285:
1280:
1279:
1258:
1253:
1252:
1231:
1226:
1225:
1206:
1205:
1183:
1182:
1160:
1159:
1137:
1136:
1115:
1107:
1106:
1085:
1077:
1076:
1055:
1047:
1046:
1024:
1023:
1001:
1000:
978:
977:
956:
951:
950:
929:
924:
923:
902:
897:
896:
877:
876:
857:
856:
837:
836:
817:
816:
797:
796:
777:
776:
757:
756:
737:
736:
717:
716:
697:
696:
686:
649:
639:
634:
633:
612:
602:
597:
596:
575:
565:
560:
559:
538:
533:
532:
511:
506:
505:
484:
479:
478:
457:
452:
451:
429:
428:
406:
405:
383:
382:
361:
356:
355:
334:
329:
328:
307:
302:
301:
280:
275:
274:
255:
254:
233:
228:
227:
206:
201:
200:
177:
176:
157:
156:
137:
136:
117:
116:
97:
96:
56:
43:
30:
25:
24:
17:
12:
11:
5:
1671:
1669:
1661:
1660:
1655:
1653:Conic sections
1650:
1640:
1639:
1635:
1634:
1618:
1603:
1573:
1542:
1529:
1514:
1496:
1472:
1455:
1453:
1450:
1445:
1441:
1437:
1433:
1429:
1428:; DB' ∩ AC = B
1425:
1417:
1413:
1409:
1353:
1350:
1337:
1317:
1292:
1288:
1265:
1261:
1238:
1234:
1213:
1193:
1190:
1170:
1167:
1147:
1144:
1122:
1118:
1114:
1092:
1088:
1084:
1062:
1058:
1054:
1034:
1031:
1011:
1008:
988:
985:
963:
959:
936:
932:
909:
905:
884:
864:
844:
824:
804:
784:
764:
744:
724:
704:
695:As above, let
685:
682:
656:
652:
646:
642:
619:
615:
609:
605:
582:
578:
572:
568:
545:
541:
518:
514:
491:
487:
464:
460:
439:
436:
416:
413:
393:
390:
368:
364:
341:
337:
314:
310:
287:
283:
262:
240:
236:
213:
209:
197:altitude lines
184:
164:
144:
124:
104:
63:
59:
55:
50:
46:
42:
37:
33:
15:
13:
10:
9:
6:
4:
3:
2:
1670:
1659:
1656:
1654:
1651:
1649:
1646:
1645:
1643:
1632:
1628:
1622:
1619:
1616:
1614:
1607:
1604:
1601:
1597:
1593:
1589:
1586:
1584:
1577:
1574:
1571:
1567:
1563:
1559:
1555:
1552:
1546:
1543:
1539:
1533:
1530:
1526:
1525:
1518:
1515:
1512:
1508:
1507:
1500:
1497:
1494:
1490:
1486:
1482:
1476:
1473:
1469:
1463:
1461:
1457:
1451:
1449:
1432:; DC' ∩ AB= C
1423:
1407:
1403:
1399:
1395:
1391:
1383:
1379:
1377:
1373:
1369:
1365:
1361:
1359:
1358:Dao Thanh Oai
1349:
1335:
1315:
1306:
1290:
1286:
1263:
1259:
1236:
1232:
1211:
1191:
1188:
1168:
1165:
1145:
1142:
1120:
1116:
1112:
1090:
1086:
1082:
1060:
1056:
1052:
1032:
1029:
1009:
1006:
986:
983:
961:
957:
934:
930:
907:
903:
882:
862:
842:
822:
802:
782:
762:
742:
722:
702:
693:
691:
681:
679:
674:
672:
654:
650:
644:
640:
617:
613:
607:
603:
580:
576:
570:
566:
543:
539:
516:
512:
489:
485:
462:
458:
437:
434:
414:
411:
391:
388:
366:
362:
339:
335:
312:
308:
285:
281:
260:
238:
234:
211:
207:
198:
182:
162:
142:
122:
102:
93:
91:
87:
83:
61:
57:
53:
48:
44:
40:
35:
31:
21:
1631:Cut-the-Knot
1621:
1612:
1606:
1582:
1576:
1561:
1545:
1537:
1532:
1522:
1517:
1504:
1499:
1484:
1475:
1467:
1389:
1388:
1375:
1371:
1367:
1363:
1362:
1355:
1307:
694:
687:
675:
94:
85:
79:
90:orthocenter
1642:Categories
1452:References
1400:P on the
1376:BC, CA, AB
175:, and let
1600:2284-5569
1570:2284-5569
1511:Mathworld
1493:1534-1178
671:collinear
1588:Archived
1554:Archived
1538:Mathesis
1436:. Then A
1396:S and a
775:. Let
273:. Let
199:. Let
1598:
1568:
1491:
1392:Let a
1278:, and
1181:, and
1105:, and
1022:, and
949:, and
895:. Let
855:, and
835:, and
755:, and
632:, and
504:, and
427:, and
327:, and
155:, and
84:, the
1422:polar
1406:lines
1402:plane
1398:point
1394:conic
1596:ISSN
1566:ISSN
1489:ISSN
669:are
226:and
95:Let
1560:."
1509:at
1483:".
1444:, C
1440:, B
1416:, d
1412:, d
80:In
1644::
1629:,
1594:,
1459:^
1348:.
1251:,
1158:,
1075:,
999:,
922:,
815:,
735:,
692:.
673:.
595:,
477:,
404:,
300:,
135:,
1446:0
1442:0
1438:0
1434:0
1430:0
1426:0
1418:c
1414:b
1410:a
1408:d
1372:P
1368:P
1336:T
1316:P
1291:1
1287:C
1264:1
1260:B
1237:1
1233:A
1212:L
1192:C
1189:P
1169:B
1166:P
1146:A
1143:P
1121:1
1117:C
1113:P
1091:1
1087:B
1083:P
1061:1
1057:A
1053:P
1033:B
1030:A
1010:A
1007:C
987:C
984:B
962:1
958:C
935:1
931:B
908:1
904:A
883:P
863:L
843:C
823:B
803:A
783:P
763:C
743:B
723:A
703:T
655:2
651:C
645:1
641:C
618:2
614:B
608:1
604:B
581:2
577:A
571:1
567:A
544:2
540:L
517:2
513:C
490:2
486:B
463:2
459:A
438:B
435:A
415:A
412:C
392:C
389:B
367:1
363:L
340:1
336:C
313:1
309:B
286:1
282:A
261:H
239:2
235:L
212:1
208:L
183:H
163:C
143:B
123:A
103:T
62:0
58:C
54:,
49:0
45:B
41:,
36:0
32:A
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