842:
73:
682:
59:
who discovered its existence in 1916. Later it was learned that the point had appeared in an article by
Christopher Zeeman published during 1899 – 1902. From 2003 onwards the Encyclopedia of Triangle Centers has been referring to this point as
1472:
1502:
1592:
1565:
1533:
1646:
1433:
1408:
44:
1678:
1428:
1617:
1660:
1643:
1359:
1111:
841:
735:
444:
56:
681:
1476:
1385:
1301:
1283:
1265:
1247:
1187:
1055:
1040:
1015:
1000:
975:
960:
743:
52:
1629:
1657:
1510:
1625:
1622:, International Journal of Computer Discovered Mathematics, Vol.1, (2016), Issue 3, page 76-79
850:
1596:
1569:
1443:
40:
29:
36:
1537:
1672:
72:
1130:
108:
1448:
1389:
1375:
1145:
766:
188:
1438:
1400:
868:
861:
32:
17:
713:. The blue 'inverted' triangle is the Gossard triangle of triangle
1366:. Dao Thanh Oai's result is generalization of all results above.
423:
Any triangle and its
Gossard triangle have the same Euler line.
726:
The construction yielding the
Gossard triangle of a triangle
829:
and its sidelines are parallel to the sidelines of triangle
809:
be the triangle formed by the Euler lines of the triangles
257:
being the intersection of the Euler lines of the triangles
238:
be the triangle formed by the Euler lines of the triangles
265:, and similarly for the other two vertices. The triangle
1620:
A generalization of the Zeeman-Gossard perspector theorem
336:
are concurrent. The point of concurrence is called the
1118:
and its sides are parallel to the sides of triangle
420:
Any triangle and its
Gossard triangle are congruent.
1327:cyclically. Then the triangle formed by the lines
1378:, Dao's result is the Gossard perspector theorem.
1392:, Dao's result is the Zeeman's generalization.
845:Paul Yiu's generalisation of Gossard triangle.
1644:CĂ©sar Eliud Lozada, Preamble before X(63787)
1362:, and the homothetic center lies on the line
1202:be two points in the plane, and let the line
8:
1403:, Dao's result is the Yiu's generalization.
758:This result is due to Christopher Zeeman.
1658:Vladimir Shelomovskii, Gossard perspector
1559:
1557:
1555:
315:be its Gossard triangle. Then the lines
1639:
1637:
840:
730:can be generalised to produce triangles
680:
71:
1466:
1464:
1460:
1186:The theorem was further generalized by
405:are respectively parallel to the lines
28:) is a special point associated with a
1503:"X(402) = Zeemann--Gossard perspector"
447:of the Gossard perspector of triangle
1496:
1494:
709:whatever be the position of the line
7:
1069:be the triangle formed by the lines
368:be the Gossard triangle of triangle
909:Let the centroids of the triangles
39:and it is designated as X(402) in
14:
860:be any point in the plane of the
426:The Gossard triangle of triangle
1647:Encyclopedia of Triangle Centers
1507:Encyclopedia of Triangle Centers
1434:Encyclopedia of Triangle Centers
1409:Encyclopedia of Triangle Centers
45:Encyclopedia of Triangle Centers
1181:
849:This generalisation is due to
705:remains congruent to triangle
689:is the Euler line of triangle
430:is the reflection of triangle
1:
746:to the sidelines of triangle
765:be any line parallel to the
697:moves parallel to the line
143:are centroids of triangles
1695:
1566:"Hyacinthos Message #7564"
434:in the Gossard perspector.
1593:"Hyacithos Message #9666"
1564:Hatzipolakis, Antreas P.
1407:The homothetic center in
825:is congruent to triangle
187:be any triangle. Let the
62:Zeeman–Gossard perspector
26:Zeeman–Gossard perspector
1236:be two points such that
777:intersect the sidelines
742:and whose sidelines are
300:be any triangle and let
1534:"Harry Clinton Gossard"
754:Zeeman’s Generalization
55:in 1998 in honour of
1661:Art of Problem Solving
846:
718:
175:
47:. The point was named
1413:Dao-Zeeman perspector
1032:be a point such that
992:be a point such that
952:be a point such that
844:
684:
445:trilinear coordinates
439:Trilinear coordinates
75:
57:Harry Clinton Gossard
1473:"Gossard Perspector"
1182:Dao's Generalisation
837:Yiu’s Generalization
1532:Kimberling, Clark.
1501:Kimberling, Clark.
1471:Kimberling, Clark.
1358:are homothetic and
1194:be a triangle. Let
1156:coincides with the
1144:coincides with the
1129:coincides with the
882:meet the sidelines
867:different from its
195:meet the sidelines
1599:on January 5, 2013
1572:on January 5, 2013
1225:respectively. Let
1106:Then the triangle
847:
805:respectively. Let
719:
338:Gossard perspector
292:Gossard perspector
223:respectively. Let
176:
49:Gossard perspector
22:Gossard perspector
1591:Grinberg, Darij.
1439:Triangle centroid
24:(also called the
1686:
1679:Triangle centers
1663:
1655:
1649:
1641:
1632:
1615:
1609:
1608:
1606:
1604:
1595:. Archived from
1588:
1582:
1581:
1579:
1577:
1568:. Archived from
1561:
1550:
1549:
1547:
1545:
1536:. Archived from
1529:
1523:
1522:
1520:
1518:
1513:on 19 April 2012
1509:. Archived from
1498:
1489:
1488:
1486:
1484:
1475:. Archived from
1468:
1444:Central triangle
1158:Gossard triangle
821:. Then triangle
282:Gossard triangle
179:Gossard triangle
41:Clark Kimberling
1694:
1693:
1689:
1688:
1687:
1685:
1684:
1683:
1669:
1668:
1667:
1666:
1656:
1652:
1642:
1635:
1618:Dao Thanh Oai,
1616:
1612:
1602:
1600:
1590:
1589:
1585:
1575:
1573:
1563:
1562:
1553:
1543:
1541:
1531:
1530:
1526:
1516:
1514:
1500:
1499:
1492:
1482:
1480:
1470:
1469:
1462:
1457:
1425:
1352:
1348:
1344:
1340:
1336:
1332:
1325:
1321:
1317:
1313:
1299:
1295:
1281:
1277:
1263:
1259:
1245:
1241:
1234:
1230:
1223:
1219:
1215:
1184:
1172:
1168:
1164:
1152:. The triangle
1100:
1096:
1089:
1085:
1078:
1074:
1052:
1037:
1030:
1012:
997:
990:
972:
957:
950:
940:
933:
926:
839:
756:
724:
722:Generalizations
701:. The triangle
685:In the figure,
441:
403:
399:
392:
388:
381:
377:
366:
362:
358:
350:
334:
327:
320:
313:
309:
305:
294:
278:
274:
270:
255:
236:
232:
228:
181:
172:
168:
164:
141:
134:
127:
120:
105:
98:
91:
84:
70:
37:triangle center
12:
11:
5:
1692:
1690:
1682:
1681:
1671:
1670:
1665:
1664:
1650:
1633:
1610:
1583:
1551:
1540:on 22 May 2013
1524:
1490:
1479:on 10 May 2012
1459:
1458:
1456:
1453:
1452:
1451:
1446:
1441:
1436:
1431:
1424:
1421:
1405:
1404:
1393:
1379:
1350:
1346:
1342:
1338:
1334:
1330:
1323:
1319:
1315:
1311:
1297:
1293:
1279:
1275:
1261:
1257:
1243:
1239:
1232:
1228:
1221:
1217:
1213:
1183:
1180:
1170:
1166:
1162:
1140:then the line
1104:
1103:
1098:
1094:
1087:
1083:
1076:
1072:
1063:
1050:
1035:
1028:
1023:
1010:
995:
988:
983:
970:
955:
948:
943:
938:
931:
924:
907:
838:
835:
755:
752:
723:
720:
679:
678:
612:
611:
563:
562:
505:
504:
440:
437:
436:
435:
424:
421:
418:
401:
397:
390:
386:
379:
375:
364:
360:
356:
349:
346:
332:
325:
318:
311:
307:
303:
293:
290:
280:is called the
276:
272:
268:
253:
234:
230:
226:
180:
177:
170:
166:
162:
139:
132:
125:
118:
103:
96:
89:
82:
69:
66:
13:
10:
9:
6:
4:
3:
2:
1691:
1680:
1677:
1676:
1674:
1662:
1659:
1654:
1651:
1648:
1645:
1640:
1638:
1634:
1631:
1627:
1623:
1621:
1614:
1611:
1598:
1594:
1587:
1584:
1571:
1567:
1560:
1558:
1556:
1552:
1539:
1535:
1528:
1525:
1512:
1508:
1504:
1497:
1495:
1491:
1478:
1474:
1467:
1465:
1461:
1454:
1450:
1447:
1445:
1442:
1440:
1437:
1435:
1432:
1430:
1427:
1426:
1422:
1420:
1418:
1414:
1410:
1402:
1398:
1394:
1391:
1387:
1384:
1380:
1377:
1373:
1369:
1368:
1367:
1365:
1361:
1357:
1354:and triangle
1353:
1326:
1307:
1303:
1300:
1289:
1285:
1282:
1271:
1267:
1264:
1253:
1249:
1246:
1235:
1224:
1209:
1205:
1201:
1197:
1193:
1189:
1188:Dao Thanh Oai
1179:
1177:
1173:
1159:
1155:
1151:
1147:
1143:
1139:
1135:
1132:
1128:
1123:
1121:
1117:
1113:
1109:
1101:
1090:
1079:
1068:
1064:
1061:
1057:
1053:
1046:
1042:
1038:
1031:
1024:
1021:
1017:
1013:
1006:
1002:
998:
991:
984:
981:
977:
973:
966:
962:
958:
951:
944:
942:respectively.
941:
934:
927:
920:
916:
912:
908:
906:respectively.
905:
901:
897:
893:
889:
885:
881:
878:Let the line
877:
876:
875:
873:
870:
866:
863:
859:
854:
852:
843:
836:
834:
832:
828:
824:
820:
816:
812:
808:
804:
800:
796:
792:
788:
784:
780:
776:
772:
768:
764:
759:
753:
751:
749:
745:
741:
737:
733:
729:
721:
716:
712:
708:
704:
700:
696:
692:
688:
683:
676:
672:
668:
664:
660:
656:
652:
648:
644:
640:
636:
632:
628:
624:
620:
617:
616:
615:
609:
605:
601:
598:
594:
591:
587:
583:
579:
575:
571:
568:
567:
566:
561:
557:
553:
549:
545:
541:
537:
533:
529:
525:
521:
517:
513:
510:
509:
508:
502:
498:
494:
490:
486:
482:
478:
474:
470:
466:
462:
458:
454:
453:
452:
450:
446:
438:
433:
429:
425:
422:
419:
416:
412:
408:
404:
393:
382:
371:
367:
352:
351:
347:
345:
343:
339:
335:
328:
321:
314:
299:
291:
289:
287:
283:
279:
264:
260:
256:
250:, the vertex
249:
245:
241:
237:
222:
218:
214:
210:
206:
202:
198:
194:
190:
186:
178:
174:respectively.
173:
158:
154:
150:
146:
142:
135:
128:
121:
114:
110:
106:
99:
92:
85:
78:
74:
67:
65:
63:
58:
54:
50:
46:
42:
38:
34:
31:
27:
23:
19:
1653:
1619:
1613:
1601:. Retrieved
1597:the original
1586:
1574:. Retrieved
1570:the original
1542:. Retrieved
1538:the original
1527:
1515:. Retrieved
1511:the original
1506:
1481:. Retrieved
1477:the original
1429:Central line
1416:
1415:of the line
1412:
1406:
1396:
1382:
1371:
1363:
1355:
1328:
1309:
1305:
1291:
1287:
1273:
1269:
1255:
1251:
1237:
1226:
1211:
1207:
1203:
1199:
1195:
1191:
1185:
1175:
1174:of triangle
1160:
1157:
1154:A'B'C'
1153:
1149:
1148:of triangle
1141:
1137:
1136:of triangle
1133:
1126:
1124:
1119:
1115:
1114:to triangle
1108:A'B'C'
1107:
1105:
1092:
1081:
1070:
1067:A'B'C'
1066:
1059:
1048:
1044:
1033:
1026:
1019:
1008:
1004:
993:
986:
979:
968:
964:
953:
946:
936:
929:
922:
918:
914:
910:
903:
899:
895:
891:
887:
883:
879:
871:
864:
857:
855:
848:
830:
826:
823:A'B'C'
822:
818:
814:
810:
807:A'B'C'
806:
802:
798:
794:
790:
789:of triangle
786:
782:
778:
774:
770:
769:of triangle
762:
760:
757:
747:
739:
738:to triangle
732:A'B'C'
731:
727:
725:
714:
710:
706:
703:A'B'C'
702:
698:
694:
690:
686:
674:
670:
666:
662:
658:
654:
650:
646:
642:
638:
634:
630:
626:
622:
618:
613:
607:
603:
599:
596:
592:
589:
585:
581:
577:
573:
569:
564:
559:
555:
551:
547:
543:
539:
535:
531:
527:
523:
519:
515:
511:
506:
500:
496:
492:
488:
484:
480:
476:
472:
468:
464:
460:
456:
448:
442:
431:
427:
414:
410:
406:
395:
384:
373:
372:. The lines
369:
354:
341:
340:of triangle
337:
330:
323:
316:
301:
297:
295:
285:
284:of triangle
281:
266:
262:
258:
251:
247:
243:
239:
224:
220:
216:
212:
208:
207:of triangle
204:
200:
196:
192:
191:of triangle
184:
182:
160:
156:
152:
148:
144:
137:
130:
123:
116:
112:
109:orthocenters
101:
94:
87:
80:
76:
61:
48:
25:
21:
15:
1131:orthocenter
693:. The line
53:John Conway
35:. It is a
1455:References
1449:Euler line
1390:Euler line
1376:Euler line
1208:BC, CA, AB
1146:Euler line
767:Euler line
734:which are
602:− (
348:Properties
189:Euler line
68:Definition
1630:2367-7775
1360:congruent
1308:. Define
1112:congruent
736:congruent
487:) :
471:) :
1673:Category
1423:See also
1401:centroid
1386:parallel
1302:parallel
1284:parallel
1266:parallel
1248:parallel
1056:parallel
1041:parallel
1016:parallel
1001:parallel
976:parallel
961:parallel
869:centroid
862:triangle
851:Paul Yiu
744:parallel
673:−
665:−
657:−
637:−
606:−
595:−
588:−
33:triangle
18:geometry
1603:18 June
1576:17 June
1544:17 June
1517:17 June
1483:17 June
1399:is the
1388:to the
1374:is the
1628:
1411:named
1206:meets
1190:. Let
773:. Let
669:) + (
565:where
507:where
111:, and
1395:When
1381:When
1370:When
1231:and A
1125:When
661:) ( 2
584:) = 2
30:plane
1626:ISSN
1605:2012
1578:2012
1546:2012
1519:2012
1485:2012
1272:and
1198:and
1091:and
1065:Let
1047:and
1025:Let
1007:and
985:Let
967:and
945:Let
935:and
917:and
902:and
890:and
856:Let
817:and
761:Let
649:) +
633:) =
614:and
558:) /
526:) =
451:are
443:The
413:and
394:and
353:Let
329:and
296:Let
288:.
261:and
246:and
219:and
203:and
183:Let
107:are
20:the
1419:.
1356:ABC
1345:, C
1337:, B
1322:, C
1318:, C
1314:, B
1304:to
1286:to
1268:to
1250:to
1220:, C
1216:, B
1210:at
1192:ABC
1176:ABC
1150:ABC
1138:ABC
1120:ABC
1116:ABC
1110:is
1058:to
1054:is
1043:to
1039:is
1018:to
1014:is
1003:to
999:is
978:to
974:is
963:to
959:is
921:be
919:CXY
915:BZX
911:AYZ
894:at
865:ABC
831:ABC
827:ABC
819:CXY
815:BZX
811:AYZ
793:at
791:ABC
771:ABC
748:ABC
740:ABC
728:ABC
715:ABC
711:XYZ
707:ABC
699:DEF
695:XYZ
691:ABC
687:DEF
653:( 2
503:) )
449:ABC
432:ABC
428:ABC
370:ABC
342:ABC
298:ABC
286:ABC
263:CDE
259:BFD
248:CDE
244:BFD
240:AEF
211:at
209:ABC
193:ABC
185:ABC
157:CDE
153:BFD
149:AEF
145:ABC
51:by
43:'s
16:In
1675::
1636:^
1624:,
1554:^
1505:.
1493:^
1463:^
1417:OH
1383:PQ
1372:HO
1364:OH
1306:CO
1290:,
1288:BO
1270:CH
1254:,
1252:BH
1204:HO
1178:.
1142:PG
1122:.
1080:,
1060:AP
1049:YP
1045:BP
1034:XP
1020:CP
1009:XP
1005:AP
994:ZP
980:BP
969:ZP
965:CP
954:YP
928:,
913:,
898:,
892:AB
888:CA
886:,
884:BC
880:PG
874:.
853:.
833:.
813:,
801:,
797:,
787:AB
785:,
783:CA
781:,
779:BC
750:.
677:)
645:+
641:(
629:,
625:,
621:(
580:,
576:,
572:(
554:,
550:,
546:(
542:)
538:,
534:,
530:(
522:,
518:,
514:(
499:,
495:,
491:(
483:,
479:,
475:(
467:,
463:,
459:(
455:(
415:AB
411:CA
409:,
407:BC
383:,
344:.
331:CC
324:BB
322:,
317:AA
242:,
215:,
205:AB
201:CA
199:,
197:BC
159:,
155:,
151:,
147:,
136:,
129:,
122:,
115:,
100:,
93:,
86:,
79:,
64:.
1607:.
1580:.
1548:.
1521:.
1487:.
1397:P
1351:O
1349:C
1347:H
1343:O
1341:B
1339:H
1335:O
1333:A
1331:H
1329:A
1324:O
1320:H
1316:O
1312:H
1310:B
1298:O
1296:A
1294:0
1292:B
1280:O
1278:A
1276:0
1274:C
1262:H
1260:A
1258:0
1256:B
1244:H
1242:A
1240:0
1238:C
1233:O
1229:H
1227:A
1222:0
1218:0
1214:0
1212:A
1200:O
1196:H
1171:g
1169:C
1167:g
1165:B
1163:g
1161:A
1134:H
1127:P
1102:.
1099:c
1097:P
1095:c
1093:G
1088:b
1086:P
1084:b
1082:G
1077:a
1075:P
1073:a
1071:G
1062:.
1051:c
1036:c
1029:c
1027:P
1022:.
1011:b
996:b
989:b
987:P
982:.
971:a
956:a
949:a
947:P
939:c
937:G
932:b
930:G
925:a
923:G
904:Z
900:Y
896:X
872:G
858:P
803:Z
799:Y
795:X
775:l
763:l
717:.
675:c
671:b
667:b
663:c
659:c
655:b
651:a
647:c
643:b
639:a
635:a
631:c
627:b
623:a
619:y
610:)
608:c
604:b
600:c
597:a
593:b
590:a
586:a
582:c
578:b
574:a
570:p
560:a
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548:a
544:y
540:c
536:b
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520:b
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512:f
501:b
497:a
493:c
489:f
485:a
481:c
477:b
473:f
469:c
465:b
461:a
457:f
417:.
402:g
400:B
398:g
396:A
391:g
389:A
387:g
385:C
380:g
378:C
376:g
374:B
365:g
363:C
361:g
359:B
357:g
355:A
333:g
326:g
319:g
312:g
310:C
308:g
306:B
304:g
302:A
277:g
275:C
273:g
271:B
269:g
267:A
254:g
252:A
235:g
233:C
231:g
229:B
227:g
225:A
221:F
217:E
213:D
171:g
169:C
167:g
165:B
163:g
161:A
140:g
138:G
133:C
131:G
126:B
124:G
119:A
117:G
113:G
104:g
102:H
97:C
95:H
90:B
88:H
83:A
81:H
77:H
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