267:% Combination of 3 figures in 1. box(0,0,10,24) draw(segment(point(0,9),point(10,9))) draw(segment(point(0,16),point(10,16))) % Proof that the Fermat point lies within the triangle A = point(7,21) B = point(1,18) C = point(9,18) D = point(4,17) E = point(7.5,23) thickness(2) draw(line(B,C)) draw(line(A,B)) draw(line(A,C)) thickness(0.5) draw(segment(D,A)) draw(segment(D,B)) draw(segment(D,C)) draw(segment(E,A)) draw(segment(E,B)) draw(segment(E,C)) label(A,0.2,170:) label(B,0.2,110:) label(C,0.2,45:) draw($ P'$ ,intersection(line(A,D),line(B,C)),0.1,280:) draw($ P$ ,D,0.1,270:) draw("\textbf{A-zone only}",D,2,350:) draw($ P'$ ,A,0.2,355:) draw($ P$ ,E,0.2,0:) draw("\textbf{B-zone and C-zone}",E,0.4,90:) draw("\textbf{C-zone only}",segment(A,B),2,120:) % Case 1 - an angle at least 120 degrees H = point(6,11.5) I = point(1,10) J = point(9,10) K = point(4.8,10.5) I H F equilateral K H G equilateral thickness(2) draw(H,I,J) thickness(0.5) draw(H,K,G) draw(segment(F,H)) draw(segment(F,I)) draw(segment(F,G)) draw(segment(K,I)) draw(segment(K,J)) draw($ A$ ,H,0.2,20:) draw($ B$ ,I,0.2,180:) draw($ C$ ,J,0.2,0:) draw($ P$ ,K,0.1,270:) draw($ Q$ ,G,0.1,210:) draw($ F$ ,F,0.2,180:) draw("\textbf{\Huge Case 1}",F,4.5,0:) mark(segment(K,I),simple,0.7) mark(segment(G,F),simple,0.7) mark(segment(H,K),double,0.7) mark(segment(K,G),double,0.7) mark(segment(G,H),double,0.7) mark(segment(H,I),triple,0.7) mark(segment(I,F),triple,0.7) mark(segment(F,H),triple,0.7) % Case 2 - no angle greater than 120 degrees L = point(4,8) M = point(3.5,5) N = point(9,5) P = point(6.2,6.1) M L O equilateral N P Q equilateral N M R equilateral S = intersection(line(N,O),line(L,R)) N S T equilateral thickness(2) draw(L,M,N) thickness(0.5) draw(N,P,Q) draw(segment(M,R)) draw(segment(N,R)) draw(segment(O,L)) draw(segment(O,M)) draw(segment(R,Q)) draw(segment(P,L)) draw(segment(P,N)) draw(segment(M,P)) draw(segment(S,M)) draw($ A$ ,L,0.2,20:) draw($ B$ ,M,0.2,180:) draw($ C$ ,N,0.2,0:) draw($ P$ ,P,0.2,180:) draw($ Q$ ,Q,0.1,180:) draw($ F$ ,O,0.2,180:) draw($ D$ ,R,0.2,0:) draw($ P_0$ ,S,0.1,45:) draw($ Q_0$ ,T,0.1,0:) draw("\textbf{\Huge Case 2}",L,2.5,0:) draw(segment(L,R)) draw(segment(N,O)) draw(segment(N,T)) mark(segment(M,P),simple,0.7) mark(segment(Q,R),simple,0.7) mark(segment(N,P),triple,0.7) mark(segment(N,Q),triple,0.7) mark(segment(P,Q),triple,0.7) mark(segment(M,N),double,0.7) mark(segment(M,R),double,0.7) mark(segment(N,R),double,0.7)
589:"}},"text\/plain":{"en":{"":"inception"}}},"{\"value\":{\"time\":\"+2008-08-31T00:00:00Z\",\"timezone\":0,\"before\":0,\"after\":0,\"precision\":11,\"calendarmodel\":\"http:\\\/\\\/www.wikidata.org\\\/entity\\\/Q1985727\"},\"type\":\"time\"}":{"text\/html":{"en":{"P571":"31 August 2008"}},"text\/plain":{"en":{"P571":"31 August 2008"}}}}": -->
369:
484:"}},"text\/plain":{"en":{"P6216":"copyrighted, dedicated to the public domain by copyright holder"}}},"{\"value\":{\"entity-type\":\"item\",\"numeric-id\":50423863,\"id\":\"Q50423863\"},\"type\":\"wikibase-entityid\"}":{"text\/html":{"en":{"P6216":"
300:
94:
381:"}},"text\/plain":{"en":{"P2093":"MRFS","P4174":"MRFS"}}},"{\"value\":{\"entity-type\":\"property\",\"numeric-id\":4174,\"id\":\"P4174\"},\"type\":\"wikibase-entityid\"}":{"text\/html":{"en":{"":"
545:"}},"text\/plain":{"en":{"":"copyright license"}}},"{\"value\":{\"entity-type\":\"item\",\"numeric-id\":98592850,\"id\":\"Q98592850\"},\"type\":\"wikibase-entityid\"}":{"text\/html":{"en":{"P275":"
480:"}},"text\/plain":{"en":{"":"copyright status"}}},"{\"value\":{\"entity-type\":\"item\",\"numeric-id\":88088423,\"id\":\"Q88088423\"},\"type\":\"wikibase-entityid\"}":{"text\/html":{"en":{"P6216":"
627:"}},"text\/plain":{"en":{"":"source of file"}}},"{\"value\":{\"entity-type\":\"item\",\"numeric-id\":66458942,\"id\":\"Q66458942\"},\"type\":\"wikibase-entityid\"}":{"text\/html":{"en":{"P7482":"
385:"}},"text\/plain":{"en":{"":"Wikimedia username"}}},"{\"value\":{\"entity-type\":\"property\",\"numeric-id\":2699,\"id\":\"P2699\"},\"type\":\"wikibase-entityid\"}":{"text\/html":{"en":{"":"
373:"}},"text\/plain":{"en":{"":"creator"}}},"{\"value\":{\"entity-type\":\"property\",\"numeric-id\":2093,\"id\":\"P2093\"},\"type\":\"wikibase-entityid\"}":{"text\/html":{"en":{"":"
623:
769:
799:
709:
389:"}},"text\/plain":{"en":{"":"URL"}}},"{\"value\":\"http:\\\/\\\/commons.wikimedia.org\\\/wiki\\\/User:MRFS\",\"type\":\"string\"}":{"text\/html":{"en":{"P2699":"
377:"}},"text\/plain":{"en":{"":"author name string"}}},"{\"value\":\"MRFS\",\"type\":\"string\"}":{"text\/html":{"en":{"P2093":"MRFS","P4174":"
258:
The main reason for this diagram is to emphasise that the Fermat point is NOT the same as the first isogonic center which is (unfortunately) a very widespread misconception. The source is licensed under the same license as the image. Feel free to edit, fix, or improve it!
378:
628:
546:
485:
481:
390:
624:
477:
386:
382:
374:
586:
542:
370:
549:"}},"text\/plain":{"en":{"P275":"released into the public domain by the copyright holder"}}}}": -->
393:"}},"text\/plain":{"en":{"P2699":"http:\/\/commons.wikimedia.org\/wiki\/User:MRFS"}}}}": -->
816:{{Information |Description={{en|1=Proof that the Fermat point cannot lie outside the triangle}} |Source=Own work by uploader |Author=
654:
572:
528:
511:
68:
64:
60:
56:
52:
48:
42:
105:
638:
495:
455:
440:
427:
862:
631:"}},"text\/plain":{"en":{"P7482":"original creation by uploader"}}}}": -->
77:
596:
556:
400:
359:
798:
793:
768:
763:
738:
733:
820:|Date=31 August 2008 |Permission= |other_versions= }} <!--{{ImageUpload|full}}--> [[Category:Fermat po
447:
488:"}},"text\/plain":{"en":{"P6216":"copyrighted"}}}}": -->
830:
The following pages on the
English Knowledge use this file (pages on other projects are not listed):
113:
244:
462:
8:
708:
703:
37:
786:{{Information |Description= |Source= |Date= |Author= |Permission= |other_versions= }}
756:{{Information |Description= |Source= |Date= |Author= |Permission= |other_versions= }}
726:{{Information |Description= |Source= |Date= |Author= |Permission= |other_versions= }}
307:
100:
349:
834:
126:
648:
606:
566:
522:
505:
410:
233:
817:
809:
779:
749:
719:
322:, without any conditions, unless such conditions are required by law.
127:
676:
Click on a date/time to view the file as it appeared at that time.
305:
I, the copyright holder of this work, release this work into the
512:
copyrighted, dedicated to the public domain by copyright holder
482:
copyrighted, dedicated to the public domain by copyright holder
852:
314:
In some countries this may not be legally possible; if so:
213:
88:
82:(SVG file, nominally 358 × 860 pixels, file size: 80 KB)
573:
released into the public domain by the copyright holder
547:
released into the public domain by the copyright holder
343:
Add a one-line explanation of what this file represents
828:
843:
674:
112:
Commons is a freely licensed media file repository.
204:A purely geometric treatment of the Fermat point
391:http:\/\/commons.wikimedia.org\/wiki\/User:MRFS
93:
209:
191:
239:
229:
219:
14:
159:
41:Size of this PNG preview of this SVG file:
318:I grant anyone the right to use this work
845:The following other wikis use this file:
262:
133:
463:http://commons.wikimedia.org/User:MRFS
299:
243:Derivative works of this file:
124:
294:
667:
13:
348:
342:
198:
128:
125:
75:
876:
347:
129:
829:
333:
298:
92:
31:
21:
330:
199:
840:
671:
329:
171:
26:
1:
655:original creation by uploader
629:original creation by uploader
825:
355:Items portrayed in this file
270:
139:
7:
149:
10:
881:
859:Usage on mk.wikipedia.org
849:Usage on es.wikipedia.org
188:
183:
668:
311:. This applies worldwide.
253:
223:
16:
764:16:34, 23 September 2008
734:08:57, 24 September 2008
704:10:20, 24 September 2008
332:
99:This is a file from the
36:
815:
785:
755:
725:
212:
103:. Information from its
263:fermat_point_scope.euk
194:Fermat Point Scope.svg
164:fermat_point_scope.euk
106:description page there
794:15:05, 31 August 2008
47:Other resolutions:
441:Wikimedia username
428:author name string
383:Wikimedia username
375:author name string
65:852 × 2,048 pixels
61:426 × 1,024 pixels
841:Global file usage
823:
557:copyright license
543:copyright license
341:
327:
326:
315:
293:
292:
250:
249:
134:
121:
120:
101:Wikimedia Commons
32:Global file usage
872:
806:
776:
746:
716:
678:
496:copyright status
478:copyright status
465:
460:
450:
445:
435:
432:
339:
313:
302:
301:
295:
289:
286:
283:
280:
275:
274:
245:Fermat proof.png
225:
215:
203:
195:
189:
177:
174:
165:
162:
155:
152:
145:
142:
135:
117:
96:
95:
89:
83:
72:
69:358 × 860 pixels
57:319 × 768 pixels
53:200 × 480 pixels
49:100 × 240 pixels
43:249 × 598 pixels
880:
879:
875:
874:
873:
871:
870:
869:
853:Punto de Fermat
839:
831:
824:
812:
804:
782:
774:
752:
744:
722:
714:
670:
669:
666:
665:
664:
663:
662:
661:
660:
659:
657:
645:
644:
643:
641:
622:
621:
620:
619:
618:
617:
616:
615:
613:
603:
602:
601:
599:
584:
583:
582:
581:
580:
579:
578:
577:
575:
563:
562:
561:
559:
540:
539:
538:
537:
536:
535:
534:
533:
531:
520:
519:
518:
517:
516:
514:
502:
501:
500:
498:
475:
474:
473:
472:
471:
470:
469:
468:
467:
466:
461:
458:
452:
451:
446:
443:
437:
436:
433:
430:
421:
420:
419:
417:
407:
406:
405:
403:
368:
367:
366:
365:
364:
362:
346:
345:
344:
320:for any purpose
287:
284:
281:
278:
273:
268:
265:
256:
251:
205:
193:
186:
181:
175:
172:
163:
160:
153:
150:
143:
140:
136:
132:
123:
122:
111:
110:
109:is shown below.
85:
81:
74:
73:
46:
12:
11:
5:
878:
868:
867:
866:
865:
863:Точка на Ферма
857:
856:
855:
842:
838:
837:
827:
822:
821:
814:
810:
807:
801:
796:
791:
788:
787:
784:
780:
777:
771:
766:
761:
758:
757:
754:
750:
747:
741:
736:
731:
728:
727:
724:
720:
717:
711:
706:
701:
697:
696:
693:
690:
687:
684:
681:
673:
658:
653:
651:
650:
649:
647:
646:
642:
639:source of file
637:
635:
634:
633:
632:
630:
626:
625:source of file
614:
612:31 August 2008
611:
609:
608:
607:
605:
604:
600:
595:
593:
592:
591:
590:
588:
576:
571:
569:
568:
567:
565:
564:
560:
555:
553:
552:
551:
550:
548:
544:
532:
527:
525:
524:
523:
521:
515:
510:
508:
507:
506:
504:
503:
499:
494:
492:
491:
490:
489:
487:
483:
479:
454:
453:
439:
438:
426:
425:
424:
423:
422:
418:
415:
413:
412:
411:
409:
408:
404:
399:
397:
396:
395:
394:
392:
388:
384:
380:
376:
372:
363:
358:
356:
353:
352:
351:
350:
338:
337:
334:
331:
325:
324:
303:
291:
290:
272:
269:
266:
264:
261:
255:
252:
248:
247:
241:
240:Other versions
237:
236:
231:
227:
226:
221:
217:
216:
214:31 August 2008
211:
207:
206:
196:
187:
185:
182:
180:
179:
169:
168:
167:
147:
131:
119:
118:
97:
87:
86:
40:
35:
34:
29:
24:
19:
9:
6:
4:
3:
2:
877:
864:
861:
860:
858:
854:
851:
850:
848:
847:
846:
836:
833:
832:
819:
813:
808:
802:
800:
797:
795:
792:
790:
789:
783:
778:
772:
770:
767:
765:
762:
760:
759:
753:
748:
742:
740:
737:
735:
732:
730:
729:
723:
718:
712:
710:
707:
705:
702:
699:
698:
694:
691:
688:
685:
682:
680:
679:
677:
656:
652:
640:
636:
610:
598:
594:
574:
570:
558:
554:
530:
526:
513:
509:
497:
493:
464:
457:
449:
442:
429:
414:
402:
398:
361:
357:
354:
335:
328:
323:
321:
316:
312:
310:
309:
308:public domain
304:
297:
296:
282:Public domain
279:Public domain
277:
276:
260:
246:
242:
238:
235:
232:
228:
222:
218:
208:
202:
197:
190:
178:
170:
166:
158:
157:
156:
148:
146:
138:
137:
115:
108:
107:
102:
98:
91:
90:
84:
79:
78:Original file
70:
66:
62:
58:
54:
50:
44:
39:
33:
30:
28:
25:
23:
20:
18:
15:
844:
835:Fermat point
743:744 × 1,053
675:
672:File history
319:
317:
306:
257:
200:
114:You can help
104:
76:
22:File history
529:copyrighted
486:copyrighted
192:Description
826:File usage
803:356 × 286
773:356 × 853
713:358 × 860
689:Dimensions
416:some value
27:File usage
686:Thumbnail
683:Date/Time
597:inception
587:inception
271:Licensing
201:English:
176:Licensing
336:Captions
224:Own work
130:Contents
805:(13 KB)
775:(40 KB)
745:(79 KB)
715:(80 KB)
700:current
695:Comment
401:creator
371:creator
360:depicts
340:English
184:Summary
144:Summary
80:
254:Source
230:Author
220:Source
154:Source
288:false
285:false
818:MRFS
811:MRFS
781:MRFS
751:MRFS
721:MRFS
692:User
448:MRFS
434:MRFS
379:MRFS
234:MRFS
210:Date
17:File
456:URL
387:URL
161:2.1
459::
444::
431::
67:|
63:|
59:|
55:|
51:|
45:.
173:3
151:2
141:1
116:.
71:.
