3094:
2700:
4463:
3089:{\displaystyle {\begin{aligned}|\mathbf {a} |&=\mathbf {a\cdot i} =(\mathbf {a} -\mathbf {x} )\mathbf {\,\cdot \,i} +\mathbf {x\cdot i} \leq |\mathbf {a} -\mathbf {x} |+\mathbf {x\cdot i} ,\\|\mathbf {b} |&=\mathbf {b\cdot j} =(\mathbf {b} -\mathbf {x} )\mathbf {\,\cdot \,j} +\mathbf {x\cdot j} \leq |\mathbf {b} -\mathbf {x} |+\mathbf {x\cdot j} ,\\|\mathbf {c} |&=\mathbf {c\cdot k} =(\mathbf {c} -\mathbf {x} )\mathbf {\,\cdot \,k} +\mathbf {x\cdot k} \leq |\mathbf {c} -\mathbf {x} |+\mathbf {x\cdot k} .\end{aligned}}}
501:
138:
5037:
4766:
20:
4787:
4516:
3301:
4392:
4243:
5237:
5032:{\displaystyle {\begin{aligned}&\csc \left(A-{\tfrac {\pi }{3}}\right):\csc \left(B-{\tfrac {\pi }{3}}\right):\csc \left(C-{\tfrac {\pi }{3}}\right)\\&=\sec \left(A+{\tfrac {\pi }{6}}\right):\sec \left(B+{\tfrac {\pi }{6}}\right):\sec \left(C+{\tfrac {\pi }{6}}\right).\end{aligned}}}
4761:{\displaystyle {\begin{aligned}&\csc \left(A+{\tfrac {\pi }{3}}\right):\csc \left(B+{\tfrac {\pi }{3}}\right):\csc \left(C+{\tfrac {\pi }{3}}\right)\\&=\sec \left(A-{\tfrac {\pi }{6}}\right):\sec \left(B-{\tfrac {\pi }{6}}\right):\sec \left(C-{\tfrac {\pi }{6}}\right).\end{aligned}}}
3489:
3119:
5567:
5415:
4258:
2642:
3958:
955:
715:
A key result that will be used is the dogleg rule, which asserts that if a triangle and a polygon have one side in common and the rest of the triangle lies inside the polygon then the triangle has a shorter perimeter than the polygon:
3810:
5054:
3348:
3296:{\displaystyle |\mathbf {a} |+|\mathbf {b} |+|\mathbf {c} |\leq |\mathbf {a} -\mathbf {x} |+|\mathbf {b} -\mathbf {x} |+|\mathbf {c} -\mathbf {x} |+\mathbf {x} \cdot (\mathbf {i} +\mathbf {j} +\mathbf {k} ).}
5444:
5292:
5757:. He solved the problem in a similar way to Fermat's, albeit using the intersection of the circumcircles of the three regular triangles instead. His pupil, Viviani, published the solution in 1659.
4387:{\displaystyle {\tfrac {\partial L}{\partial x}},{\tfrac {\partial L}{\partial y}},{\tfrac {\partial L}{\partial z}},{\tfrac {\partial L}{\partial \alpha }},{\tfrac {\partial L}{\partial \beta }}}
4792:
4521:
2705:
2001:
3629:
119:
For each isosceles triangle draw a circle, in each case with center on the new vertex of the isosceles triangle and with radius equal to each of the two new sides of that isosceles triangle.
2552:
2459:
1081:
2315:
1536:
6064:
606:
1673:
1366:
1221:
2403:
2354:
1624:
1575:
1138:
4238:{\displaystyle L=x+y+z+\lambda _{1}(x^{2}+y^{2}-2xy\cos(\alpha )-a^{2})+\lambda _{2}(y^{2}+z^{2}-2yz\cos(\beta )-b^{2})+\lambda _{3}(z^{2}+x^{2}-2zx\cos(\alpha +\beta )-c^{2})}
2510:
666:
1451:
2197:
1769:
1717:
1495:
702:
2230:
483:. So, the lines joining the centers of the circles also intersect at 60° angles. Therefore, the centers of the circles form an equilateral triangle. This is known as
1017:
1795:
745:
47:, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible or, equivalently, the
5737:. However these different names can be confusing and are perhaps best avoided. The problem is that much of the literature blurs the distinction between the
968:. Associate each vertex with its remote zone; that is, the half-plane beyond the (extended) opposite side. These 3 zones cover the entire plane except for
5232:{\displaystyle 1-u+uvw\sec \left(A-{\tfrac {\pi }{6}}\right):1-v+uvw\sec \left(B-{\tfrac {\pi }{6}}\right):1-w+uvw\sec \left(C-{\tfrac {\pi }{6}}\right)}
3740:
3815:
the third inequality still holds, the other two inequalities are unchanged. The proof now continues as above (adding the three inequalities and using
3484:{\displaystyle |\mathbf {a} |+|\mathbf {b} |+|\mathbf {c} |\leq |\mathbf {a} -\mathbf {x} |+|\mathbf {b} -\mathbf {x} |+|\mathbf {c} -\mathbf {x} |}
116:, with base the side in question, 30-degree angles at the base, and the third vertex of each isosceles triangle lying outside the original triangle.
6165:
6160:
6134:
6071:
23:
Fig 1. Construction of the first isogonic center, X(13). When no angle of the triangle exceeds 120°, this point is the Fermat point.
5929:
5910:
5891:
5872:
5562:{\displaystyle \sin \left(A-{\tfrac {\pi }{3}}\right):\sin \left(B-{\tfrac {\pi }{3}}\right):\sin \left(C-{\tfrac {\pi }{3}}\right).}
5410:{\displaystyle \sin \left(A+{\tfrac {\pi }{3}}\right):\sin \left(B+{\tfrac {\pi }{3}}\right):\sin \left(C+{\tfrac {\pi }{3}}\right).}
6129:
129:
In what follows "Case 1" means the triangle has an angle exceeding 120°. "Case 2" means no angle of the triangle exceeds 120°.
6170:
1863:
6144:
6028:
3506:
6189:
6114:
2637:{\displaystyle {\overrightarrow {OA}},\ {\overrightarrow {OB}},\ {\overrightarrow {OC}},\ {\overrightarrow {OX}}}
6005:
6015:
2247:
1500:
522:
6010:
1285:
2408:
1022:
6109:
126:
When a triangle has an angle greater than 120°, the Fermat point is sited at the obtuse-angled vertex.
2464:
620:
6212:
6057:
3861:
2027:
intersect. This point is commonly called the first isogonic center. Carry out the same exercise with
1386:
3856:
Another approach to finding the point within a triangle, from which the sum of the distances to the
1629:
1174:
6094:
2359:
2328:
2147:
1580:
1549:
1094:
1730:
1678:
1456:
671:
5769:
or Fermat–Weber point, the point minimizing the sum of distances to more than three given points.
4454:. However the elimination is a long and tedious business, and the end result covers only Case 2.
2202:
5806:
6181:
5946:
5754:
56:
449:
The lines joining the centers of the circles in Fig. 2 are perpendicular to the line segments
299:
264:
6104:
5609:
4500:
1774:
484:
194:
6034:
5964:
950:{\displaystyle {\text{perimeter}}>|AB|+|AX|+|XB|=|AB|+|AC|+|CX|+|XB|\geq |AB|+|AC|+|BC|.}
5850:
5698:
3865:
91:
67:
5926:
5907:
5888:
5869:
991:
122:
The intersection inside the original triangle between the two circles is the Fermat point.
8:
6038:
5772:
5672:
3948:. Using the method of Lagrange multipliers we have to find the minimum of the Lagrangian
737:
6139:
6119:
5838:
113:
28:
4462:
5977:
5668:
3857:
708:
will denote the points inside the triangle and will be taken to include its boundary
98:
6080:
5834:
5830:
5818:
5766:
5428:
5276:
3805:{\displaystyle |\mathbf {0} |\leq |\mathbf {0} -\mathbf {x} |+\mathbf {x\cdot k} ,}
355:
238:
168:
63:
52:
48:
5980:
6044:
6023:
5933:
5914:
5895:
5876:
5846:
5782:
5777:
307:
224:
5959:
Christopher J. Bradley and Geoff C. Smith, "The locations of triangle centers",
5578:
5248:
3875:
We draw lines from the point within the triangle to its vertices and call them
3869:
51:
of the three vertices. It is so named because this problem was first raised by
6206:
6099:
5787:
4467:
78:
The Fermat point of a triangle with largest angle at most 120° is simply its
6175:
5664:
4490:
704:
If such a point exists then it will be the Fermat point. In what follows
5842:
5821:(1994). "Central Points and Central Lines in the Plane of a Triangle".
5645:
3674:
meet at angles of 120°. Nevertheless, it is easily fixed by redefining
5985:
500:
463:. For example, the line joining the center of the circle containing
137:
5745:
whereas it is only in Case 2 above that they are actually the same.
6049:
36:
4493:
of the three constructed equilateral triangles are concurrent at
4486:(13) are all equal to 120° (Case 2), or 60°, 60°, 120° (Case 1).
4475:
When the largest angle of the triangle is not larger than 120°,
1264:
is equilateral and has the orientation shown. Then the triangle
359:
19:
94:
on each of two arbitrarily chosen sides of the given triangle.
2535:
the Fermat point is coincident with the first isogonic center
5701:
punctured at its own center, and could be any point therein.
6041:
Interactive sketch generalizes the Fermat-Torricelli point.
16:
Triangle center minimizing sum of distances to each vertex
5965:
http://forumgeom.fau.edu/FG2006volume6/FG200607index.html
1282:
so these two triangles are congruent and it follows that
5753:
This question was proposed by Fermat, as a challenge to
4466:
The two isogonic centers are the intersections of three
197:
because the second is a 60° rotation of the first about
5975:
5540:
5503:
5466:
5388:
5351:
5314:
5213:
5155:
5097:
5006:
4969:
4932:
4888:
4851:
4814:
4774:
Trilinear coordinates for the second isogonic center,
4735:
4698:
4661:
4617:
4580:
4543:
4470:
whose paired vertices are the vertices of the triangle
4363:
4338:
4313:
4288:
4263:
1230:
Without loss of generality, suppose that the angle at
391:. However it is easily modified to cover Case 1. Then
112:
On each of two arbitrarily chosen sides, construct an
5648:. The three lines meet at the Euler infinity point,
5447:
5295:
5057:
4790:
4519:
4482:
The angles subtended by the sides of the triangle at
4261:
3961:
3743:
3509:
3351:
3122:
2703:
2555:
2467:
2411:
2362:
2331:
2250:
2205:
2150:
1866:
1777:
1733:
1681:
1632:
1583:
1552:
1503:
1459:
1389:
1383:, by the dogleg rule the length of this path exceeds
1288:
1177:
1097:
1025:
994:
748:
674:
623:
525:
5807:
3648:This argument fails when the triangle has an angle
5561:
5409:
5231:
5031:
4760:
4386:
4237:
3804:
3623:
3483:
3295:
3088:
2636:
2549:be any five points in a plane. Denote the vectors
2504:
2453:
2397:
2348:
2309:
2224:
2191:
1996:{\displaystyle d(P)=|PA|+|PB|+|PC|=|AP|+|PQ|+|QD|}
1995:
1789:
1763:
1711:
1667:
1618:
1569:
1530:
1489:
1445:
1360:
1215:
1132:
1075:
1011:
949:
696:
660:
600:
3860:of the triangle is minimal, is to use one of the
3624:{\displaystyle |OA|+|OB|+|OC|\leq |XA|+|XB|+|XC|}
1807:the Fermat point lies at the obtuse-angled vertex
6204:
160:attached to the sides of the arbitrary triangle
101:to the opposite vertex of the original triangle.
608:The aim of this section is to identify a point
490:
241:(they lie on a circle). Similarly, the points
141:Fig 2. Geometry of the first isogonic center.
4255:Equating each of the five partial derivatives
4252:are the lengths of the sides of the triangle.
1234:is ≥ 120°. Construct the equilateral triangle
976:clearly lies in either one or two of them. If
6065:
1813:Case 2. The triangle has no angle ≥ 120°.
1226:Case 1. The triangle has an angle ≥ 120°.
364:This proof applies only in Case 2, since if
104:The two lines intersect at the Fermat point.
6161:List of things named after Pierre de Fermat
5045:Trilinear coordinates for the Fermat point:
3717:because the angle between the unit vectors
6072:
6058:
5817:
3889:. Also, let the lengths of these lines be
736:. Then the polygon's perimeter is, by the
5686:(14) meets the Euler line at midpoint of
5575:The following triangles are equilateral:
3020:
3016:
2894:
2890:
2768:
2764:
1832:, and construct the equilateral triangle
383:which switches the relative positions of
62:The Fermat point gives a solution to the
4461:
499:
470:and the center of the circle containing
136:
108:An alternative method is the following:
18:
6135:Fermat's theorem on sums of two squares
6045:A practical example of the Fermat point
3851:
2006:which is simply the length of the path
1368:which is simply the length of the path
495:
145:Fig. 2 shows the equilateral triangles
6205:
6166:Wiles's proof of Fermat's Last Theorem
5953:
5944:
2310:{\displaystyle d(P_{0})=|AD|<d(P).}
358:(they intersect at a single point).
179:and cut one another at angles of 60°.
167:. Here is a proof using properties of
6053:
5976:
3864:methods; specifically, the method of
1531:{\displaystyle P\in \Delta ,P\neq A.}
1083:by the dogleg rule. Alternatively if
504:Fig 3. Geometry of the Fermat point
6130:Fermat's theorem (stationary points)
6079:
3893:respectively. Let the angle between
3833:) to reach the same conclusion that
601:{\displaystyle d(P)=|PA|+|PB|+|PC|.}
175:in Fig 2 all intersect at the point
132:
5729:respectively. Alternatives are the
1817:Construct the equilateral triangle
1361:{\displaystyle d(P)=|CP|+|PQ|+|QF|}
86:, which is constructed as follows:
13:
6029:The Wolfram Demonstrations Project
5947:"Encyclopedia of Triangle Centers"
5938:
5697:The Fermat point lies in the open
4374:
4366:
4349:
4341:
4324:
4316:
4299:
4291:
4274:
4266:
2540:
2454:{\displaystyle d(P_{0})\leq d(P')}
2343:
1564:
1510:
1271:is a 60° rotation of the triangle
1076:{\displaystyle d(P')=d(A)<d(P)}
988:zones’ intersection) then setting
477:, is perpendicular to the segment
14:
6224:
5998:
6171:Fermat's Last Theorem in fiction
6035:Fermat-Torricelli generalization
5927:Encyclopedia of Triangle Centers
5908:Encyclopedia of Triangle Centers
5889:Encyclopedia of Triangle Centers
5870:Encyclopedia of Triangle Centers
3795:
3789:
3776:
3768:
3750:
3472:
3464:
3446:
3438:
3420:
3412:
3394:
3376:
3358:
3283:
3275:
3267:
3256:
3243:
3235:
3217:
3209:
3191:
3183:
3165:
3147:
3129:
3075:
3069:
3056:
3048:
3035:
3029:
3021:
3008:
3000:
2989:
2983:
2966:
2949:
2943:
2930:
2922:
2909:
2903:
2895:
2882:
2874:
2863:
2857:
2840:
2823:
2817:
2804:
2796:
2783:
2777:
2769:
2756:
2748:
2737:
2731:
2714:
2505:{\displaystyle d(P_{0})<d(P)}
732:to cut the polygon at the point
661:{\displaystyle d(P_{0})<d(P)}
376:lies inside the circumcircle of
6145:Fermat's right triangle theorem
6115:Fermat polygonal number theorem
4503:for the first isogonic center,
1446:{\displaystyle |AC|+|AF|=d(A).}
302:, this implies that the points
73:
5969:
5919:
5900:
5881:
5862:
5835:10.1080/0025570X.1994.11996210
5811:
5800:
4232:
4216:
4204:
4160:
4144:
4128:
4122:
4078:
4062:
4046:
4040:
3996:
3841:) must be the Fermat point of
3781:
3763:
3755:
3745:
3617:
3606:
3598:
3587:
3579:
3568:
3560:
3549:
3541:
3530:
3522:
3511:
3477:
3459:
3451:
3433:
3425:
3407:
3399:
3389:
3381:
3371:
3363:
3353:
3287:
3263:
3248:
3230:
3222:
3204:
3196:
3178:
3170:
3160:
3152:
3142:
3134:
3124:
3061:
3043:
3012:
2996:
2971:
2961:
2935:
2917:
2886:
2870:
2845:
2835:
2809:
2791:
2760:
2744:
2719:
2709:
2499:
2493:
2484:
2471:
2448:
2437:
2428:
2415:
2392:
2386:
2377:
2366:
2301:
2295:
2285:
2274:
2267:
2254:
2182:
2171:
2154:
1989:
1978:
1970:
1959:
1951:
1940:
1932:
1921:
1913:
1902:
1894:
1883:
1876:
1870:
1758:
1752:
1743:
1737:
1706:
1700:
1691:
1685:
1668:{\displaystyle d(A)\leq d(P')}
1662:
1651:
1642:
1636:
1613:
1607:
1598:
1587:
1484:
1478:
1469:
1463:
1437:
1431:
1421:
1410:
1402:
1391:
1354:
1343:
1335:
1324:
1316:
1305:
1298:
1292:
1216:{\displaystyle d(P')<d(P).}
1207:
1201:
1192:
1181:
1127:
1121:
1112:
1101:
1070:
1064:
1055:
1049:
1040:
1029:
940:
929:
921:
910:
902:
891:
883:
872:
864:
853:
845:
834:
826:
815:
807:
796:
788:
777:
769:
758:
655:
649:
640:
627:
591:
580:
572:
561:
553:
542:
535:
529:
1:
5793:
4457:
2398:{\displaystyle d(P')<d(P)}
2349:{\displaystyle P'\in \Omega }
2192:{\displaystyle d(P_{0}=|AD|.}
2049:. By the angular restriction
1619:{\displaystyle d(P')<d(P)}
1570:{\displaystyle P'\in \Omega }
1376:is constrained to lie within
1133:{\displaystyle d(P')<d(P)}
1087:is in only one zone, say the
508:Given any Euclidean triangle
306:are concyclic. So, using the
171:to show that the three lines
1764:{\displaystyle d(A)<d(P)}
1712:{\displaystyle d(A)<d(P)}
1490:{\displaystyle d(A)<d(P)}
697:{\displaystyle P\neq P_{0}.}
491:Location of the Fermat point
7:
6011:Encyclopedia of Mathematics
6006:"Fermat-Torricelli problem"
5760:
5721:(14) are also known as the
3734:which exceeds 120°. Since
2225:{\displaystyle P\neq P_{0}}
726:is the common side, extend
10:
6229:
5748:
5705:
5423:The isogonal conjugate of
5271:The isogonal conjugate of
3656:because there is no point
97:Draw a line from each new
6153:
6087:
6039:Dynamic Geometry Sketches
5644:(16) are parallel to the
5619:Circumcevian triangle of
4479:(13) is the Fermat point.
3952:, which is expressed as:
3925:. Then the angle between
3862:mathematical optimization
2676:be the unit vectors from
344:lies on the line segment
223:. By the converse of the
5247:respectively denote the
4394:to zero and eliminating
6110:Fermat's little theorem
5932:April 19, 2012, at the
5913:April 19, 2012, at the
5894:April 19, 2012, at the
5875:April 19, 2012, at the
5429:second isodynamic point
3638:is the Fermat point of
3324:at angles of 120° then
2529:is the Fermat point of
1801:is the Fermat point of
1790:{\displaystyle P\neq A}
1144:is the intersection of
515:and an arbitrary point
310:applied to the segment
308:inscribed angle theorem
300:inscribed angle theorem
265:inscribed angle theorem
227:applied to the segment
225:inscribed angle theorem
55:in a private letter to
45:Fermat–Torricelli point
6193:(popular science book)
5755:Evangelista Torricelli
5588:Antipedal triangle of
5563:
5411:
5277:first isodynamic point
5233:
5033:
4762:
4471:
4388:
4239:
3806:
3625:
3485:
3297:
3090:
2662:respectively, and let
2638:
2506:
2455:
2399:
2350:
2311:
2226:
2193:
2144:is a straight line so
2092:must lie somewhere on
1997:
1791:
1765:
1713:
1669:
1620:
1571:
1532:
1491:
1447:
1362:
1217:
1134:
1077:
1013:
951:
698:
662:
602:
505:
142:
57:Evangelista Torricelli
24:
6190:Fermat's Last Theorem
6095:Fermat's Last Theorem
5735:negative Fermat point
5731:positive Fermat point
5610:Circumcevian triangle
5564:
5412:
5234:
5034:
4763:
4501:Trilinear coordinates
4465:
4389:
4240:
3807:
3626:
3486:
3298:
3091:
2639:
2507:
2456:
2400:
2351:
2325:. From above a point
2312:
2227:
2194:
2072:is a 60° rotation of
2040:, and find the point
1998:
1846:is a 60° rotation of
1792:
1766:
1714:
1670:
1621:
1572:
1546:. From above a point
1533:
1492:
1448:
1363:
1257:so that the triangle
1218:
1162:there exists a point
1135:
1078:
1014:
964:be any point outside
952:
699:
663:
603:
503:
140:
80:first isogonic center
68:Steiner tree problems
22:
5823:Mathematics Magazine
5699:orthocentroidal disk
5445:
5293:
5055:
4788:
4517:
4259:
3959:
3866:Lagrange multipliers
3852:Lagrange multipliers
3741:
3507:
3349:
3120:
2701:
2553:
2465:
2409:
2360:
2329:
2248:
2203:
2148:
1864:
1828:be any point inside
1775:
1731:
1679:
1630:
1581:
1550:
1501:
1457:
1387:
1286:
1175:
1095:
1023:
1012:{\displaystyle P'=A}
992:
746:
672:
621:
523:
496:Traditional geometry
92:equilateral triangle
6182:Fermat's Last Tango
5963:6 (2006), 57--70.
5961:Forum Geometricorum
5945:Kimberling, Clark.
5925:Entry X(16) in the
5906:Entry X(15) in the
5887:Entry X(14) in the
5868:Entry X(13) in the
5727:second Fermat point
3500:. In other words,
2019:be the point where
980:is in two (say the
738:triangle inequality
6120:Fermat pseudoprime
6105:Fermat's principle
6027:by Chris Boucher,
5978:Weisstein, Eric W.
5743:first Fermat point
5723:first Fermat point
5602:Pedal triangle of
5595:Pedal triangle of
5579:antipedal triangle
5559:
5549:
5512:
5475:
5407:
5397:
5360:
5323:
5229:
5222:
5164:
5106:
5029:
5027:
5015:
4978:
4941:
4897:
4860:
4823:
4758:
4756:
4744:
4707:
4670:
4626:
4589:
4552:
4472:
4384:
4382:
4357:
4332:
4307:
4282:
4235:
3802:
3621:
3481:
3293:
3086:
3084:
2634:
2533:. In other words,
2502:
2451:
2395:
2346:
2307:
2222:
2189:
1993:
1805:. In other words,
1787:
1761:
1709:
1665:
1616:
1567:
1528:
1487:
1443:
1358:
1253:itself) construct
1241:and for any point
1213:
1130:
1073:
1009:
947:
694:
658:
598:
506:
485:Napoleon's Theorem
143:
114:isosceles triangle
70:for three points.
39:, also called the
29:Euclidean geometry
25:
6200:
6199:
5819:Kimberling, Clark
5669:nine-point center
5548:
5511:
5474:
5396:
5359:
5322:
5249:Boolean variables
5221:
5163:
5105:
5014:
4977:
4940:
4896:
4859:
4822:
4743:
4706:
4669:
4625:
4588:
4551:
4417:eventually gives
4381:
4356:
4331:
4306:
4281:
3837:(or in this case
2632:
2618:
2611:
2597:
2590:
2576:
2569:
2356:exists such that
2321:to range outside
1797:which means that
1577:exists such that
1542:to range outside
752:
350:. So, the lines
133:Location of X(13)
59:, who solved it.
6220:
6213:Triangle centers
6081:Pierre de Fermat
6074:
6067:
6060:
6051:
6050:
6019:
5992:
5991:
5990:
5973:
5967:
5957:
5951:
5950:
5942:
5936:
5923:
5917:
5904:
5898:
5885:
5879:
5866:
5860:
5854:
5815:
5809:
5804:
5773:Lester's theorem
5767:Geometric median
5712:isogonic centers
5568:
5566:
5565:
5560:
5555:
5551:
5550:
5541:
5518:
5514:
5513:
5504:
5481:
5477:
5476:
5467:
5416:
5414:
5413:
5408:
5403:
5399:
5398:
5389:
5366:
5362:
5361:
5352:
5329:
5325:
5324:
5315:
5265:
5246:
5238:
5236:
5235:
5230:
5228:
5224:
5223:
5214:
5170:
5166:
5165:
5156:
5112:
5108:
5107:
5098:
5038:
5036:
5035:
5030:
5028:
5021:
5017:
5016:
5007:
4984:
4980:
4979:
4970:
4947:
4943:
4942:
4933:
4907:
4903:
4899:
4898:
4889:
4866:
4862:
4861:
4852:
4829:
4825:
4824:
4815:
4794:
4767:
4765:
4764:
4759:
4757:
4750:
4746:
4745:
4736:
4713:
4709:
4708:
4699:
4676:
4672:
4671:
4662:
4636:
4632:
4628:
4627:
4618:
4595:
4591:
4590:
4581:
4558:
4554:
4553:
4544:
4523:
4453:
4442:
4427:
4416:
4393:
4391:
4390:
4385:
4383:
4380:
4372:
4364:
4358:
4355:
4347:
4339:
4333:
4330:
4322:
4314:
4308:
4305:
4297:
4289:
4283:
4280:
4272:
4264:
4251:
4244:
4242:
4241:
4236:
4231:
4230:
4185:
4184:
4172:
4171:
4159:
4158:
4143:
4142:
4103:
4102:
4090:
4089:
4077:
4076:
4061:
4060:
4021:
4020:
4008:
4007:
3995:
3994:
3951:
3947:
3936:
3930:
3924:
3920:
3914:
3908:
3904:
3898:
3892:
3888:
3847:
3840:
3836:
3832:
3811:
3809:
3808:
3803:
3798:
3784:
3779:
3771:
3766:
3758:
3753:
3748:
3733:
3726:
3716:
3714:
3706:
3696:
3692:
3688:
3673:
3659:
3655:
3644:
3637:
3630:
3628:
3627:
3622:
3620:
3609:
3601:
3590:
3582:
3571:
3563:
3552:
3544:
3533:
3525:
3514:
3499:
3490:
3488:
3487:
3482:
3480:
3475:
3467:
3462:
3454:
3449:
3441:
3436:
3428:
3423:
3415:
3410:
3402:
3397:
3392:
3384:
3379:
3374:
3366:
3361:
3356:
3341:
3323:
3319:
3302:
3300:
3299:
3294:
3286:
3278:
3270:
3259:
3251:
3246:
3238:
3233:
3225:
3220:
3212:
3207:
3199:
3194:
3186:
3181:
3173:
3168:
3163:
3155:
3150:
3145:
3137:
3132:
3127:
3112:
3095:
3093:
3092:
3087:
3085:
3078:
3064:
3059:
3051:
3046:
3038:
3024:
3011:
3003:
2992:
2974:
2969:
2964:
2952:
2938:
2933:
2925:
2920:
2912:
2898:
2885:
2877:
2866:
2848:
2843:
2838:
2826:
2812:
2807:
2799:
2794:
2786:
2772:
2759:
2751:
2740:
2722:
2717:
2712:
2693:
2679:
2675:
2661:
2643:
2641:
2640:
2635:
2633:
2628:
2620:
2616:
2612:
2607:
2599:
2595:
2591:
2586:
2578:
2574:
2570:
2565:
2557:
2548:
2532:
2528:
2519:
2515:
2511:
2509:
2508:
2503:
2483:
2482:
2461:it follows that
2460:
2458:
2457:
2452:
2447:
2427:
2426:
2404:
2402:
2401:
2396:
2376:
2355:
2353:
2352:
2347:
2339:
2324:
2320:
2316:
2314:
2313:
2308:
2288:
2277:
2266:
2265:
2243:
2239:
2235:
2231:
2229:
2228:
2223:
2221:
2220:
2198:
2196:
2195:
2190:
2185:
2174:
2166:
2165:
2143:
2125:
2121:
2112:
2104:it follows that
2103:
2095:
2091:
2082:
2078:
2071:
2064:
2057:
2048:
2039:
2036:as you did with
2035:
2026:
2022:
2018:
2009:
2002:
2000:
1999:
1994:
1992:
1981:
1973:
1962:
1954:
1943:
1935:
1924:
1916:
1905:
1897:
1886:
1856:
1852:
1845:
1838:
1831:
1827:
1823:
1804:
1800:
1796:
1794:
1793:
1788:
1770:
1768:
1767:
1762:
1726:
1722:
1718:
1716:
1715:
1710:
1675:it follows that
1674:
1672:
1671:
1666:
1661:
1625:
1623:
1622:
1617:
1597:
1576:
1574:
1573:
1568:
1560:
1545:
1541:
1537:
1535:
1534:
1529:
1496:
1494:
1493:
1488:
1452:
1450:
1449:
1444:
1424:
1413:
1405:
1394:
1382:
1375:
1371:
1367:
1365:
1364:
1359:
1357:
1346:
1338:
1327:
1319:
1308:
1281:
1277:
1270:
1263:
1256:
1252:
1248:
1244:
1240:
1233:
1222:
1220:
1219:
1214:
1191:
1169:
1165:
1161:
1157:
1154:for every point
1151:
1147:
1143:
1139:
1137:
1136:
1131:
1111:
1090:
1086:
1082:
1080:
1079:
1074:
1039:
1018:
1016:
1015:
1010:
1002:
987:
983:
979:
975:
971:
967:
963:
956:
954:
953:
948:
943:
932:
924:
913:
905:
894:
886:
875:
867:
856:
848:
837:
829:
818:
810:
799:
791:
780:
772:
761:
753:
750:
735:
731:
730:
725:
724:
711:
707:
703:
701:
700:
695:
690:
689:
667:
665:
664:
659:
639:
638:
616:
607:
605:
604:
599:
594:
583:
575:
564:
556:
545:
518:
514:
482:
481:
476:
469:
462:
461:
457:
453:
445:
441:
437:
423:is concyclic so
422:
418:
402:
390:
386:
382:
375:
371:
353:
349:
348:
343:
339:
327:
315:
314:
305:
297:
285:
274:
262:
254:
244:
236:
232:
231:
222:
211:
200:
192:
178:
174:
169:concyclic points
166:
159:
64:geometric median
49:geometric median
41:Torricelli point
6228:
6227:
6223:
6222:
6221:
6219:
6218:
6217:
6203:
6202:
6201:
6196:
6149:
6140:Fermat's spiral
6083:
6078:
6004:
6001:
5996:
5995:
5981:"Fermat Points"
5974:
5970:
5958:
5954:
5943:
5939:
5934:Wayback Machine
5924:
5920:
5915:Wayback Machine
5905:
5901:
5896:Wayback Machine
5886:
5882:
5877:Wayback Machine
5867:
5863:
5858:
5816:
5812:
5805:
5801:
5796:
5783:Napoleon points
5778:Triangle center
5763:
5751:
5708:
5539:
5532:
5528:
5502:
5495:
5491:
5465:
5458:
5454:
5446:
5443:
5442:
5387:
5380:
5376:
5350:
5343:
5339:
5313:
5306:
5302:
5294:
5291:
5290:
5251:
5244:
5212:
5205:
5201:
5154:
5147:
5143:
5096:
5089:
5085:
5056:
5053:
5052:
5026:
5025:
5005:
4998:
4994:
4968:
4961:
4957:
4931:
4924:
4920:
4905:
4904:
4887:
4880:
4876:
4850:
4843:
4839:
4813:
4806:
4802:
4791:
4789:
4786:
4785:
4755:
4754:
4734:
4727:
4723:
4697:
4690:
4686:
4660:
4653:
4649:
4634:
4633:
4616:
4609:
4605:
4579:
4572:
4568:
4542:
4535:
4531:
4520:
4518:
4515:
4514:
4460:
4444:
4429:
4418:
4415:
4408:
4401:
4395:
4373:
4365:
4362:
4348:
4340:
4337:
4323:
4315:
4312:
4298:
4290:
4287:
4273:
4265:
4262:
4260:
4257:
4256:
4249:
4226:
4222:
4180:
4176:
4167:
4163:
4154:
4150:
4138:
4134:
4098:
4094:
4085:
4081:
4072:
4068:
4056:
4052:
4016:
4012:
4003:
3999:
3990:
3986:
3960:
3957:
3956:
3949:
3938:
3932:
3926:
3922:
3916:
3910:
3906:
3900:
3894:
3890:
3876:
3854:
3842:
3838:
3834:
3816:
3788:
3780:
3775:
3767:
3762:
3754:
3749:
3744:
3742:
3739:
3738:
3728:
3718:
3710:
3708:
3698:
3694:
3690:
3675:
3661:
3657:
3649:
3639:
3635:
3616:
3605:
3597:
3586:
3578:
3567:
3559:
3548:
3540:
3529:
3521:
3510:
3508:
3505:
3504:
3495:
3476:
3471:
3463:
3458:
3450:
3445:
3437:
3432:
3424:
3419:
3411:
3406:
3398:
3393:
3388:
3380:
3375:
3370:
3362:
3357:
3352:
3350:
3347:
3346:
3325:
3321:
3307:
3282:
3274:
3266:
3255:
3247:
3242:
3234:
3229:
3221:
3216:
3208:
3203:
3195:
3190:
3182:
3177:
3169:
3164:
3159:
3151:
3146:
3141:
3133:
3128:
3123:
3121:
3118:
3117:
3100:
3083:
3082:
3068:
3060:
3055:
3047:
3042:
3028:
3015:
3007:
2999:
2982:
2975:
2970:
2965:
2960:
2957:
2956:
2942:
2934:
2929:
2921:
2916:
2902:
2889:
2881:
2873:
2856:
2849:
2844:
2839:
2834:
2831:
2830:
2816:
2808:
2803:
2795:
2790:
2776:
2763:
2755:
2747:
2730:
2723:
2718:
2713:
2708:
2704:
2702:
2699:
2698:
2681:
2677:
2663:
2645:
2621:
2619:
2600:
2598:
2579:
2577:
2558:
2556:
2554:
2551:
2550:
2546:
2543:
2541:Vector analysis
2530:
2527:
2521:
2517:
2513:
2478:
2474:
2466:
2463:
2462:
2440:
2422:
2418:
2410:
2407:
2406:
2369:
2361:
2358:
2357:
2332:
2330:
2327:
2326:
2322:
2318:
2284:
2273:
2261:
2257:
2249:
2246:
2245:
2241:
2237:
2233:
2216:
2212:
2204:
2201:
2200:
2181:
2170:
2161:
2157:
2149:
2146:
2145:
2139:
2133:
2127:
2123:
2120:
2114:
2111:
2105:
2097:
2093:
2090:
2084:
2080:
2073:
2066:
2059:
2056:
2050:
2047:
2041:
2037:
2034:
2028:
2024:
2020:
2017:
2011:
2007:
1988:
1977:
1969:
1958:
1950:
1939:
1931:
1920:
1912:
1901:
1893:
1882:
1865:
1862:
1861:
1854:
1847:
1840:
1833:
1829:
1825:
1818:
1802:
1798:
1776:
1773:
1772:
1732:
1729:
1728:
1724:
1720:
1680:
1677:
1676:
1654:
1631:
1628:
1627:
1590:
1582:
1579:
1578:
1553:
1551:
1548:
1547:
1543:
1539:
1502:
1499:
1498:
1458:
1455:
1454:
1420:
1409:
1401:
1390:
1388:
1385:
1384:
1377:
1373:
1369:
1353:
1342:
1334:
1323:
1315:
1304:
1287:
1284:
1283:
1279:
1272:
1265:
1258:
1254:
1250:
1246:
1242:
1235:
1231:
1184:
1176:
1173:
1172:
1167:
1163:
1159:
1155:
1149:
1145:
1141:
1104:
1096:
1093:
1092:
1088:
1084:
1032:
1024:
1021:
1020:
995:
993:
990:
989:
985:
981:
977:
973:
969:
965:
961:
939:
928:
920:
909:
901:
890:
882:
871:
863:
852:
844:
833:
825:
814:
806:
795:
787:
776:
768:
757:
749:
747:
744:
743:
733:
728:
727:
722:
721:
709:
705:
685:
681:
673:
670:
669:
634:
630:
622:
619:
618:
615:
609:
590:
579:
571:
560:
552:
541:
524:
521:
520:
516:
509:
498:
493:
479:
478:
471:
464:
459:
455:
451:
450:
443:
439:
424:
420:
404:
392:
388:
384:
377:
373:
365:
351:
346:
345:
341:
329:
317:
312:
311:
303:
287:
279:
268:
256:
248:
245:are concyclic.
242:
234:
229:
228:
213:
202:
198:
183:
176:
172:
161:
146:
135:
76:
17:
12:
11:
5:
6226:
6216:
6215:
6198:
6197:
6195:
6194:
6186:
6185:(2000 musical)
6178:
6173:
6168:
6163:
6157:
6155:
6151:
6150:
6148:
6147:
6142:
6137:
6132:
6127:
6122:
6117:
6112:
6107:
6102:
6097:
6091:
6089:
6085:
6084:
6077:
6076:
6069:
6062:
6054:
6048:
6047:
6042:
6032:
6020:
6000:
5999:External links
5997:
5994:
5993:
5968:
5952:
5937:
5918:
5899:
5880:
5861:
5856:
5829:(3): 163–187.
5810:
5798:
5797:
5795:
5792:
5791:
5790:
5785:
5780:
5775:
5770:
5762:
5759:
5750:
5747:
5707:
5704:
5703:
5702:
5695:
5676:
5653:
5626:
5625:
5624:
5617:
5607:
5600:
5593:
5586:
5572:
5571:
5570:
5569:
5558:
5554:
5547:
5544:
5538:
5535:
5531:
5527:
5524:
5521:
5517:
5510:
5507:
5501:
5498:
5494:
5490:
5487:
5484:
5480:
5473:
5470:
5464:
5461:
5457:
5453:
5450:
5437:
5436:
5420:
5419:
5418:
5417:
5406:
5402:
5395:
5392:
5386:
5383:
5379:
5375:
5372:
5369:
5365:
5358:
5355:
5349:
5346:
5342:
5338:
5335:
5332:
5328:
5321:
5318:
5312:
5309:
5305:
5301:
5298:
5285:
5284:
5268:
5267:
5241:
5240:
5239:
5227:
5220:
5217:
5211:
5208:
5204:
5200:
5197:
5194:
5191:
5188:
5185:
5182:
5179:
5176:
5173:
5169:
5162:
5159:
5153:
5150:
5146:
5142:
5139:
5136:
5133:
5130:
5127:
5124:
5121:
5118:
5115:
5111:
5104:
5101:
5095:
5092:
5088:
5084:
5081:
5078:
5075:
5072:
5069:
5066:
5063:
5060:
5047:
5046:
5042:
5041:
5040:
5039:
5024:
5020:
5013:
5010:
5004:
5001:
4997:
4993:
4990:
4987:
4983:
4976:
4973:
4967:
4964:
4960:
4956:
4953:
4950:
4946:
4939:
4936:
4930:
4927:
4923:
4919:
4916:
4913:
4910:
4908:
4906:
4902:
4895:
4892:
4886:
4883:
4879:
4875:
4872:
4869:
4865:
4858:
4855:
4849:
4846:
4842:
4838:
4835:
4832:
4828:
4821:
4818:
4812:
4809:
4805:
4801:
4798:
4795:
4793:
4780:
4779:
4771:
4770:
4769:
4768:
4753:
4749:
4742:
4739:
4733:
4730:
4726:
4722:
4719:
4716:
4712:
4705:
4702:
4696:
4693:
4689:
4685:
4682:
4679:
4675:
4668:
4665:
4659:
4656:
4652:
4648:
4645:
4642:
4639:
4637:
4635:
4631:
4624:
4621:
4615:
4612:
4608:
4604:
4601:
4598:
4594:
4587:
4584:
4578:
4575:
4571:
4567:
4564:
4561:
4557:
4550:
4547:
4541:
4538:
4534:
4530:
4527:
4524:
4522:
4509:
4508:
4498:
4487:
4480:
4468:vesicae piscis
4459:
4456:
4413:
4406:
4399:
4379:
4376:
4371:
4368:
4361:
4354:
4351:
4346:
4343:
4336:
4329:
4326:
4321:
4318:
4311:
4304:
4301:
4296:
4293:
4286:
4279:
4276:
4271:
4268:
4246:
4245:
4234:
4229:
4225:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4194:
4191:
4188:
4183:
4179:
4175:
4170:
4166:
4162:
4157:
4153:
4149:
4146:
4141:
4137:
4133:
4130:
4127:
4124:
4121:
4118:
4115:
4112:
4109:
4106:
4101:
4097:
4093:
4088:
4084:
4080:
4075:
4071:
4067:
4064:
4059:
4055:
4051:
4048:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4019:
4015:
4011:
4006:
4002:
3998:
3993:
3989:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3870:law of cosines
3853:
3850:
3813:
3812:
3801:
3797:
3794:
3791:
3787:
3783:
3778:
3774:
3770:
3765:
3761:
3757:
3752:
3747:
3632:
3631:
3619:
3615:
3612:
3608:
3604:
3600:
3596:
3593:
3589:
3585:
3581:
3577:
3574:
3570:
3566:
3562:
3558:
3555:
3551:
3547:
3543:
3539:
3536:
3532:
3528:
3524:
3520:
3517:
3513:
3492:
3491:
3479:
3474:
3470:
3466:
3461:
3457:
3453:
3448:
3444:
3440:
3435:
3431:
3427:
3422:
3418:
3414:
3409:
3405:
3401:
3396:
3391:
3387:
3383:
3378:
3373:
3369:
3365:
3360:
3355:
3304:
3303:
3292:
3289:
3285:
3281:
3277:
3273:
3269:
3265:
3262:
3258:
3254:
3250:
3245:
3241:
3237:
3232:
3228:
3224:
3219:
3215:
3211:
3206:
3202:
3198:
3193:
3189:
3185:
3180:
3176:
3172:
3167:
3162:
3158:
3154:
3149:
3144:
3140:
3136:
3131:
3126:
3097:
3096:
3081:
3077:
3074:
3071:
3067:
3063:
3058:
3054:
3050:
3045:
3041:
3037:
3034:
3031:
3027:
3023:
3019:
3014:
3010:
3006:
3002:
2998:
2995:
2991:
2988:
2985:
2981:
2978:
2976:
2973:
2968:
2963:
2959:
2958:
2955:
2951:
2948:
2945:
2941:
2937:
2932:
2928:
2924:
2919:
2915:
2911:
2908:
2905:
2901:
2897:
2893:
2888:
2884:
2880:
2876:
2872:
2869:
2865:
2862:
2859:
2855:
2852:
2850:
2847:
2842:
2837:
2833:
2832:
2829:
2825:
2822:
2819:
2815:
2811:
2806:
2802:
2798:
2793:
2789:
2785:
2782:
2779:
2775:
2771:
2767:
2762:
2758:
2754:
2750:
2746:
2743:
2739:
2736:
2733:
2729:
2726:
2724:
2721:
2716:
2711:
2707:
2706:
2631:
2627:
2624:
2615:
2610:
2606:
2603:
2594:
2589:
2585:
2582:
2573:
2568:
2564:
2561:
2542:
2539:
2525:
2501:
2498:
2495:
2492:
2489:
2486:
2481:
2477:
2473:
2470:
2450:
2446:
2443:
2439:
2436:
2433:
2430:
2425:
2421:
2417:
2414:
2394:
2391:
2388:
2385:
2382:
2379:
2375:
2372:
2368:
2365:
2345:
2342:
2338:
2335:
2306:
2303:
2300:
2297:
2294:
2291:
2287:
2283:
2280:
2276:
2272:
2269:
2264:
2260:
2256:
2253:
2219:
2215:
2211:
2208:
2188:
2184:
2180:
2177:
2173:
2169:
2164:
2160:
2156:
2153:
2137:
2131:
2118:
2109:
2088:
2054:
2045:
2032:
2015:
2004:
2003:
1991:
1987:
1984:
1980:
1976:
1972:
1968:
1965:
1961:
1957:
1953:
1949:
1946:
1942:
1938:
1934:
1930:
1927:
1923:
1919:
1915:
1911:
1908:
1904:
1900:
1896:
1892:
1889:
1885:
1881:
1878:
1875:
1872:
1869:
1786:
1783:
1780:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1664:
1660:
1657:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1615:
1612:
1609:
1606:
1603:
1600:
1596:
1593:
1589:
1586:
1566:
1563:
1559:
1556:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1442:
1439:
1436:
1433:
1430:
1427:
1423:
1419:
1416:
1412:
1408:
1404:
1400:
1397:
1393:
1356:
1352:
1349:
1345:
1341:
1337:
1333:
1330:
1326:
1322:
1318:
1314:
1311:
1307:
1303:
1300:
1297:
1294:
1291:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1190:
1187:
1183:
1180:
1129:
1126:
1123:
1120:
1117:
1114:
1110:
1107:
1103:
1100:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1038:
1035:
1031:
1028:
1008:
1005:
1001:
998:
958:
957:
946:
942:
938:
935:
931:
927:
923:
919:
916:
912:
908:
904:
900:
897:
893:
889:
885:
881:
878:
874:
870:
866:
862:
859:
855:
851:
847:
843:
840:
836:
832:
828:
824:
821:
817:
813:
809:
805:
802:
798:
794:
790:
786:
783:
779:
775:
771:
767:
764:
760:
756:
741:
693:
688:
684:
680:
677:
657:
654:
651:
648:
645:
642:
637:
633:
629:
626:
613:
597:
593:
589:
586:
582:
578:
574:
570:
567:
563:
559:
555:
551:
548:
544:
540:
537:
534:
531:
528:
497:
494:
492:
489:
286:. Therefore,
267:. Similarly,
182:The triangles
134:
131:
124:
123:
120:
117:
106:
105:
102:
95:
75:
72:
15:
9:
6:
4:
3:
2:
6225:
6214:
6211:
6210:
6208:
6192:
6191:
6187:
6184:
6183:
6179:
6177:
6174:
6172:
6169:
6167:
6164:
6162:
6159:
6158:
6156:
6152:
6146:
6143:
6141:
6138:
6136:
6133:
6131:
6128:
6126:
6123:
6121:
6118:
6116:
6113:
6111:
6108:
6106:
6103:
6101:
6100:Fermat number
6098:
6096:
6093:
6092:
6090:
6086:
6082:
6075:
6070:
6068:
6063:
6061:
6056:
6055:
6052:
6046:
6043:
6040:
6036:
6033:
6030:
6026:
6025:
6021:
6017:
6013:
6012:
6007:
6003:
6002:
5988:
5987:
5982:
5979:
5972:
5966:
5962:
5956:
5948:
5941:
5935:
5931:
5928:
5922:
5916:
5912:
5909:
5903:
5897:
5893:
5890:
5884:
5878:
5874:
5871:
5865:
5852:
5848:
5844:
5840:
5836:
5832:
5828:
5824:
5820:
5814:
5808:
5803:
5799:
5789:
5788:Weber problem
5786:
5784:
5781:
5779:
5776:
5774:
5771:
5768:
5765:
5764:
5758:
5756:
5746:
5744:
5740:
5736:
5732:
5728:
5724:
5720:
5716:
5713:
5700:
5696:
5693:
5689:
5685:
5681:
5677:
5674:
5673:Lester circle
5670:
5666:
5662:
5658:
5654:
5651:
5647:
5643:
5639:
5635:
5631:
5627:
5622:
5618:
5615:
5611:
5608:
5605:
5601:
5598:
5594:
5591:
5587:
5584:
5580:
5577:
5576:
5574:
5573:
5556:
5552:
5545:
5542:
5536:
5533:
5529:
5525:
5522:
5519:
5515:
5508:
5505:
5499:
5496:
5492:
5488:
5485:
5482:
5478:
5471:
5468:
5462:
5459:
5455:
5451:
5448:
5441:
5440:
5439:
5438:
5434:
5430:
5426:
5422:
5421:
5404:
5400:
5393:
5390:
5384:
5381:
5377:
5373:
5370:
5367:
5363:
5356:
5353:
5347:
5344:
5340:
5336:
5333:
5330:
5326:
5319:
5316:
5310:
5307:
5303:
5299:
5296:
5289:
5288:
5287:
5286:
5282:
5278:
5274:
5270:
5269:
5263:
5260:< 120°), (
5259:
5256:< 120°), (
5255:
5250:
5242:
5225:
5218:
5215:
5209:
5206:
5202:
5198:
5195:
5192:
5189:
5186:
5183:
5180:
5177:
5174:
5171:
5167:
5160:
5157:
5151:
5148:
5144:
5140:
5137:
5134:
5131:
5128:
5125:
5122:
5119:
5116:
5113:
5109:
5102:
5099:
5093:
5090:
5086:
5082:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5051:
5050:
5049:
5048:
5044:
5043:
5022:
5018:
5011:
5008:
5002:
4999:
4995:
4991:
4988:
4985:
4981:
4974:
4971:
4965:
4962:
4958:
4954:
4951:
4948:
4944:
4937:
4934:
4928:
4925:
4921:
4917:
4914:
4911:
4909:
4900:
4893:
4890:
4884:
4881:
4877:
4873:
4870:
4867:
4863:
4856:
4853:
4847:
4844:
4840:
4836:
4833:
4830:
4826:
4819:
4816:
4810:
4807:
4803:
4799:
4796:
4784:
4783:
4782:
4781:
4777:
4773:
4772:
4751:
4747:
4740:
4737:
4731:
4728:
4724:
4720:
4717:
4714:
4710:
4703:
4700:
4694:
4691:
4687:
4683:
4680:
4677:
4673:
4666:
4663:
4657:
4654:
4650:
4646:
4643:
4640:
4638:
4629:
4622:
4619:
4613:
4610:
4606:
4602:
4599:
4596:
4592:
4585:
4582:
4576:
4573:
4569:
4565:
4562:
4559:
4555:
4548:
4545:
4539:
4536:
4532:
4528:
4525:
4513:
4512:
4511:
4510:
4506:
4502:
4499:
4496:
4492:
4491:circumcircles
4488:
4485:
4481:
4478:
4474:
4473:
4469:
4464:
4455:
4451:
4447:
4441:
4437:
4433:
4426:
4422:
4412:
4405:
4398:
4377:
4369:
4359:
4352:
4344:
4334:
4327:
4319:
4309:
4302:
4294:
4284:
4277:
4269:
4253:
4227:
4223:
4219:
4213:
4210:
4207:
4201:
4198:
4195:
4192:
4189:
4186:
4181:
4177:
4173:
4168:
4164:
4155:
4151:
4147:
4139:
4135:
4131:
4125:
4119:
4116:
4113:
4110:
4107:
4104:
4099:
4095:
4091:
4086:
4082:
4073:
4069:
4065:
4057:
4053:
4049:
4043:
4037:
4034:
4031:
4028:
4025:
4022:
4017:
4013:
4009:
4004:
4000:
3991:
3987:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3955:
3954:
3953:
3946:
3942:
3935:
3929:
3919:
3913:
3903:
3897:
3887:
3883:
3879:
3873:
3871:
3867:
3863:
3859:
3849:
3846:
3831:
3827:
3823:
3819:
3799:
3792:
3785:
3772:
3759:
3737:
3736:
3735:
3732:
3725:
3721:
3713:
3705:
3701:
3686:
3682:
3678:
3672:
3668:
3664:
3653:
3646:
3643:
3613:
3610:
3602:
3594:
3591:
3583:
3575:
3572:
3564:
3556:
3553:
3545:
3537:
3534:
3526:
3518:
3515:
3503:
3502:
3501:
3498:
3468:
3455:
3442:
3429:
3416:
3403:
3385:
3367:
3345:
3344:
3343:
3340:
3336:
3332:
3328:
3318:
3314:
3310:
3290:
3279:
3271:
3260:
3252:
3239:
3226:
3213:
3200:
3187:
3174:
3156:
3138:
3116:
3115:
3114:
3111:
3107:
3103:
3079:
3072:
3065:
3052:
3039:
3032:
3025:
3017:
3004:
2993:
2986:
2979:
2977:
2953:
2946:
2939:
2926:
2913:
2906:
2899:
2891:
2878:
2867:
2860:
2853:
2851:
2827:
2820:
2813:
2800:
2787:
2780:
2773:
2765:
2752:
2741:
2734:
2727:
2725:
2697:
2696:
2695:
2692:
2688:
2684:
2674:
2670:
2666:
2660:
2656:
2652:
2648:
2629:
2625:
2622:
2613:
2608:
2604:
2601:
2592:
2587:
2583:
2580:
2571:
2566:
2562:
2559:
2547:O, A, B, C, X
2538:
2536:
2524:
2520:. That means
2496:
2490:
2487:
2479:
2475:
2468:
2444:
2441:
2434:
2431:
2423:
2419:
2412:
2389:
2383:
2380:
2373:
2370:
2363:
2340:
2336:
2333:
2304:
2298:
2292:
2289:
2281:
2278:
2270:
2262:
2258:
2251:
2240:won't lie on
2217:
2213:
2209:
2206:
2199:Moreover, if
2186:
2178:
2175:
2167:
2162:
2158:
2151:
2142:
2136:
2130:
2117:
2113:lies between
2108:
2101:
2087:
2077:
2070:
2063:
2053:
2044:
2031:
2014:
1985:
1982:
1974:
1966:
1963:
1955:
1947:
1944:
1936:
1928:
1925:
1917:
1909:
1906:
1898:
1890:
1887:
1879:
1873:
1867:
1860:
1859:
1858:
1851:
1844:
1837:
1822:
1815:
1814:
1810:
1808:
1784:
1781:
1778:
1755:
1749:
1746:
1740:
1734:
1703:
1697:
1694:
1688:
1682:
1658:
1655:
1648:
1645:
1639:
1633:
1610:
1604:
1601:
1594:
1591:
1584:
1561:
1557:
1554:
1525:
1522:
1519:
1516:
1513:
1507:
1504:
1481:
1475:
1472:
1466:
1460:
1440:
1434:
1428:
1425:
1417:
1414:
1406:
1398:
1395:
1381:
1350:
1347:
1339:
1331:
1328:
1320:
1312:
1309:
1301:
1295:
1289:
1276:
1269:
1262:
1239:
1228:
1227:
1223:
1210:
1204:
1198:
1195:
1188:
1185:
1178:
1171:
1124:
1118:
1115:
1108:
1105:
1098:
1067:
1061:
1058:
1052:
1046:
1043:
1036:
1033:
1026:
1006:
1003:
999:
996:
944:
936:
933:
925:
917:
914:
906:
898:
895:
887:
879:
876:
868:
860:
857:
849:
841:
838:
830:
822:
819:
811:
803:
800:
792:
784:
781:
773:
765:
762:
754:
742:
739:
719:
718:
717:
713:
691:
686:
682:
678:
675:
652:
646:
643:
635:
631:
624:
612:
595:
587:
584:
576:
568:
565:
557:
549:
546:
538:
532:
526:
513:
502:
488:
486:
475:
468:
447:
438:. Therefore,
436:
432:
428:
416:
412:
408:
400:
396:
381:
369:
362:
361:
357:
337:
333:
325:
321:
309:
301:
298:. Using the
295:
291:
283:
276:
272:
266:
263:, using the
260:
252:
246:
240:
233:, the points
226:
221:
217:
210:
206:
196:
191:
187:
180:
170:
165:
158:
154:
150:
139:
130:
127:
121:
118:
115:
111:
110:
109:
103:
100:
96:
93:
90:Construct an
89:
88:
87:
85:
81:
71:
69:
65:
60:
58:
54:
50:
46:
42:
38:
34:
30:
21:
6188:
6180:
6176:Fermat Prize
6125:Fermat point
6124:
6024:Fermat Point
6022:
6009:
5984:
5971:
5960:
5955:
5940:
5921:
5902:
5883:
5864:
5826:
5822:
5813:
5802:
5752:
5742:
5739:Fermat point
5738:
5734:
5730:
5726:
5722:
5718:
5714:
5711:
5709:
5691:
5687:
5683:
5679:
5665:circumcenter
5660:
5656:
5649:
5641:
5637:
5633:
5629:
5620:
5613:
5603:
5596:
5589:
5582:
5432:
5427:(14) is the
5424:
5280:
5275:(13) is the
5272:
5261:
5257:
5253:
4775:
4504:
4494:
4483:
4476:
4449:
4445:
4439:
4435:
4431:
4424:
4420:
4410:
4403:
4396:
4254:
4247:
3944:
3940:
3933:
3927:
3917:
3911:
3901:
3895:
3885:
3881:
3877:
3874:
3855:
3844:
3829:
3825:
3821:
3817:
3814:
3730:
3723:
3719:
3711:
3707:. Note that
3703:
3699:
3689:and placing
3684:
3680:
3676:
3670:
3666:
3662:
3651:
3647:
3641:
3633:
3496:
3493:
3338:
3334:
3330:
3326:
3316:
3312:
3308:
3305:
3109:
3105:
3101:
3098:
2690:
2686:
2682:
2672:
2668:
2664:
2658:
2654:
2650:
2646:
2544:
2534:
2522:
2244:which means
2232:then either
2140:
2134:
2128:
2126:which means
2115:
2106:
2099:
2085:
2075:
2068:
2065:. Moreover,
2061:
2058:lies inside
2051:
2042:
2029:
2012:
2005:
1849:
1842:
1835:
1820:
1816:
1812:
1811:
1806:
1379:
1274:
1267:
1260:
1237:
1229:
1225:
1224:
1153:
1091:-zone, then
959:
714:
610:
511:
507:
473:
466:
448:
434:
430:
426:
419:which means
414:
410:
406:
398:
394:
379:
367:
363:
340:, the point
335:
331:
323:
319:
293:
289:
281:
277:
270:
258:
250:
247:
219:
215:
208:
204:
189:
185:
181:
163:
156:
152:
148:
144:
128:
125:
107:
83:
79:
77:
74:Construction
61:
44:
40:
33:Fermat point
32:
26:
5655:The points
1453:Therefore,
972:itself and
5794:References
5667:, and the
5663:(14), the
5646:Euler line
5628:The lines
5264:< 120°)
4458:Properties
4438:) = − sin
3715:| ≤ 1
3634:and hence
2317:Now allow
1538:Now allow
617:such that
356:concurrent
352:RC, BQ, AP
328:. Because
173:RC, BQ, AP
6016:EMS Press
5986:MathWorld
5859:, p. 174.
5717:(13) and
5678:The line
5671:lie on a
5636:(15) and
5543:π
5537:−
5526:
5506:π
5500:−
5489:
5469:π
5463:−
5452:
5391:π
5374:
5354:π
5337:
5317:π
5300:
5216:π
5210:−
5199:
5178:−
5158:π
5152:−
5141:
5120:−
5100:π
5094:−
5083:
5062:−
5009:π
4992:
4972:π
4955:
4935:π
4918:
4891:π
4885:−
4874:
4854:π
4848:−
4837:
4817:π
4811:−
4800:
4738:π
4732:−
4721:
4701:π
4695:−
4684:
4664:π
4658:−
4647:
4620:π
4603:
4583:π
4566:
4546:π
4529:
4378:β
4375:∂
4367:∂
4353:α
4350:∂
4342:∂
4325:∂
4317:∂
4300:∂
4292:∂
4275:∂
4267:∂
4220:−
4214:β
4208:α
4202:
4187:−
4152:λ
4132:−
4126:β
4120:
4105:−
4070:λ
4050:−
4044:α
4038:
4023:−
3988:λ
3793:⋅
3773:−
3760:≤
3654:> 120°
3565:≤
3469:−
3443:−
3417:−
3404:≤
3261:⋅
3240:−
3214:−
3188:−
3175:≤
3073:⋅
3053:−
3040:≤
3033:⋅
3018:⋅
3005:−
2987:⋅
2947:⋅
2927:−
2914:≤
2907:⋅
2892:⋅
2879:−
2861:⋅
2821:⋅
2801:−
2788:≤
2781:⋅
2766:⋅
2753:−
2735:⋅
2630:→
2609:→
2588:→
2567:→
2432:≤
2344:Ω
2341:∈
2210:≠
1782:≠
1646:≤
1565:Ω
1562:∈
1520:≠
1511:Δ
1508:∈
1170:such that
888:≥
751:perimeter
679:≠
433:= 60° = ∠
370:> 120°
239:concyclic
195:congruent
6207:Category
5930:Archived
5911:Archived
5892:Archived
5873:Archived
5761:See also
5741:and the
5733:and the
5725:and the
5690:(2) and
3868:and the
3858:vertices
3697:so that
3494:for all
3320:meet at
2516:outside
2512:for all
2445:′
2374:′
2337:′
2096:. Since
1771:for all
1723:outside
1719:for all
1659:′
1595:′
1558:′
1497:for all
1249:(except
1189:′
1158:outside
1109:′
1037:′
1019:implies
1000:′
668:for all
442:lies on
372:, point
201:. Hence
37:triangle
6154:Related
6018:, 2001
5851:1573021
5843:2690608
5749:History
5706:Aliases
5245:u, v, w
4250:a, b, c
3891:x, y, z
3113:gives
3099:Adding
2405:and as
1839:. Then
1727:. Thus
1626:and as
5849:
5841:
5659:(13),
5243:where
4452:= 120°
4423:= sin
4248:where
3709:|
3660:where
3342:, so
2680:along
2617:
2596:
2575:
2079:about
2010:. Let
1853:about
1824:, let
1278:about
1140:where
417:= 120°
403:hence
360:Q.E.D.
338:= 180°
296:= 180°
284:= 120°
273:= 120°
261:= 120°
99:vertex
53:Fermat
31:, the
5855:See X
5839:JSTOR
5652:(30).
5435:(16):
5283:(15):
4778:(14):
4507:(13):
4497:(13).
3679:= − (
2102:= 60°
2083:, so
1372:. As
1152:. So
401:= 60°
354:are
326:= 60°
255:, so
253:= 60°
84:X(13)
35:of a
6088:Work
5710:The
5694:(4).
5682:(13)
5640:(14)
5632:(13)
5623:(16)
5616:(15)
5606:(16)
5599:(15)
5592:(14)
5585:(13)
4489:The
4430:sin(
4428:and
4419:sin
3939:π −
3931:and
3915:and
3899:and
2545:Let
2488:<
2381:<
2290:<
2122:and
2023:and
2008:APQD
1857:so
1747:<
1695:<
1602:<
1473:<
1370:CPQF
1196:<
1148:and
1116:<
1059:<
984:and
960:Let
755:>
644:<
519:let
421:BPCF
387:and
368:BAC
304:BPCF
243:AFCQ
237:are
235:ARBF
212:and
193:are
66:and
6037:at
5831:doi
5612:of
5581:of
5523:sin
5486:sin
5449:sin
5371:sin
5334:sin
5297:sin
5196:sec
5138:sec
5080:sec
4989:sec
4952:sec
4915:sec
4871:csc
4834:csc
4797:csc
4718:sec
4681:sec
4644:sec
4600:csc
4563:csc
4526:csc
4443:so
4199:cos
4117:cos
4035:cos
3937:is
3921:be
3905:be
3845:ABC
3727:is
3693:at
3645:.
3642:ABC
3306:If
2644:by
2236:or
2100:CDB
2076:BDA
2069:BCF
2062:ABC
1850:CPB
1843:CQD
1836:CPQ
1821:BCD
1380:ABC
1275:AFQ
1268:ABP
1261:AQP
1245:in
1238:AFB
1166:in
720:If
512:ABC
474:AQC
467:ARB
435:BFA
431:BCP
429:= ∠
427:BFP
415:AFC
413:+ ∠
411:AFB
409:= ∠
407:BFC
399:AFC
397:= ∠
395:AFB
380:BPC
336:BFA
334:+ ∠
332:BFP
324:BCP
322:= ∠
320:BFP
294:BPC
292:+ ∠
290:BFC
282:BFC
278:So
271:AFC
259:AFB
251:ARB
220:ACF
218:= ∠
216:AQF
209:ABF
207:= ∠
205:ARF
190:BAQ
188:, △
186:RAC
164:ABC
157:CPB
155:, △
153:AQC
151:, △
149:ARB
82:or
43:or
27:In
6209::
6014:,
6008:,
5983:.
5857:13
5847:MR
5845:.
5837:.
5827:67
5825:.
5431:,
5279:,
4448:=
4434:+
4409:,
4402:,
3943:−
3909:,
3884:,
3880:,
3872:.
3848:.
3828:=
3824:+
3820:+
3722:,
3702:=
3683:+
3669:,
3665:,
3337:=
3333:+
3329:+
3315:,
3311:,
3108:,
3104:,
2694:.
2689:,
2685:,
2671:,
2667:,
2657:,
2653:,
2649:,
2537:.
2242:AD
2129:AP
2094:AD
2025:CF
2021:AD
1809:.
1164:P'
1150:BC
1146:AP
1142:P'
729:AC
723:AB
712:.
487:.
480:AP
460:CR
458:,
456:BQ
454:,
452:AP
446:.
444:FP
347:AP
316:,
313:BP
275:.
230:AF
6073:e
6066:t
6059:v
6031:.
5989:.
5949:.
5853:.
5833::
5719:X
5715:X
5692:X
5688:X
5684:X
5680:X
5675:.
5661:X
5657:X
5650:X
5642:X
5638:X
5634:X
5630:X
5621:X
5614:X
5604:X
5597:X
5590:X
5583:X
5557:.
5553:)
5546:3
5534:C
5530:(
5520::
5516:)
5509:3
5497:B
5493:(
5483::
5479:)
5472:3
5460:A
5456:(
5433:X
5425:X
5405:.
5401:)
5394:3
5385:+
5382:C
5378:(
5368::
5364:)
5357:3
5348:+
5345:B
5341:(
5331::
5327:)
5320:3
5311:+
5308:A
5304:(
5281:X
5273:X
5266:.
5262:C
5258:B
5254:A
5252:(
5226:)
5219:6
5207:C
5203:(
5193:w
5190:v
5187:u
5184:+
5181:w
5175:1
5172::
5168:)
5161:6
5149:B
5145:(
5135:w
5132:v
5129:u
5126:+
5123:v
5117:1
5114::
5110:)
5103:6
5091:A
5087:(
5077:w
5074:v
5071:u
5068:+
5065:u
5059:1
5023:.
5019:)
5012:6
5003:+
5000:C
4996:(
4986::
4982:)
4975:6
4966:+
4963:B
4959:(
4949::
4945:)
4938:6
4929:+
4926:A
4922:(
4912:=
4901:)
4894:3
4882:C
4878:(
4868::
4864:)
4857:3
4845:B
4841:(
4831::
4827:)
4820:3
4808:A
4804:(
4776:X
4752:.
4748:)
4741:6
4729:C
4725:(
4715::
4711:)
4704:6
4692:B
4688:(
4678::
4674:)
4667:6
4655:A
4651:(
4641:=
4630:)
4623:3
4614:+
4611:C
4607:(
4597::
4593:)
4586:3
4577:+
4574:B
4570:(
4560::
4556:)
4549:3
4540:+
4537:A
4533:(
4505:X
4495:X
4484:X
4477:X
4450:β
4446:α
4440:β
4436:β
4432:α
4425:β
4421:α
4414:3
4411:λ
4407:2
4404:λ
4400:1
4397:λ
4370:L
4360:,
4345:L
4335:,
4328:z
4320:L
4310:,
4303:y
4295:L
4285:,
4278:x
4270:L
4233:)
4228:2
4224:c
4217:)
4211:+
4205:(
4196:x
4193:z
4190:2
4182:2
4178:x
4174:+
4169:2
4165:z
4161:(
4156:3
4148:+
4145:)
4140:2
4136:b
4129:)
4123:(
4114:z
4111:y
4108:2
4100:2
4096:z
4092:+
4087:2
4083:y
4079:(
4074:2
4066:+
4063:)
4058:2
4054:a
4047:)
4041:(
4032:y
4029:x
4026:2
4018:2
4014:y
4010:+
4005:2
4001:x
3997:(
3992:1
3984:+
3981:z
3978:+
3975:y
3972:+
3969:x
3966:=
3963:L
3950:L
3945:β
3941:α
3934:Z
3928:X
3923:β
3918:Z
3912:Y
3907:α
3902:Y
3896:X
3886:Z
3882:Y
3878:X
3843:△
3839:C
3835:O
3830:0
3826:k
3822:j
3818:i
3800:,
3796:k
3790:x
3786:+
3782:|
3777:x
3769:0
3764:|
3756:|
3751:0
3746:|
3731:C
3729:∠
3724:j
3720:i
3712:k
3704:0
3700:c
3695:C
3691:O
3687:)
3685:j
3681:i
3677:k
3671:c
3667:b
3663:a
3658:O
3652:C
3650:∠
3640:△
3636:O
3618:|
3614:C
3611:X
3607:|
3603:+
3599:|
3595:B
3592:X
3588:|
3584:+
3580:|
3576:A
3573:X
3569:|
3561:|
3557:C
3554:O
3550:|
3546:+
3542:|
3538:B
3535:O
3531:|
3527:+
3523:|
3519:A
3516:O
3512:|
3497:x
3478:|
3473:x
3465:c
3460:|
3456:+
3452:|
3447:x
3439:b
3434:|
3430:+
3426:|
3421:x
3413:a
3408:|
3400:|
3395:c
3390:|
3386:+
3382:|
3377:b
3372:|
3368:+
3364:|
3359:a
3354:|
3339:0
3335:k
3331:j
3327:i
3322:O
3317:c
3313:b
3309:a
3291:.
3288:)
3284:k
3280:+
3276:j
3272:+
3268:i
3264:(
3257:x
3253:+
3249:|
3244:x
3236:c
3231:|
3227:+
3223:|
3218:x
3210:b
3205:|
3201:+
3197:|
3192:x
3184:a
3179:|
3171:|
3166:c
3161:|
3157:+
3153:|
3148:b
3143:|
3139:+
3135:|
3130:a
3125:|
3110:c
3106:b
3102:a
3080:.
3076:k
3070:x
3066:+
3062:|
3057:x
3049:c
3044:|
3036:k
3030:x
3026:+
3022:k
3013:)
3009:x
3001:c
2997:(
2994:=
2990:k
2984:c
2980:=
2972:|
2967:c
2962:|
2954:,
2950:j
2944:x
2940:+
2936:|
2931:x
2923:b
2918:|
2910:j
2904:x
2900:+
2896:j
2887:)
2883:x
2875:b
2871:(
2868:=
2864:j
2858:b
2854:=
2846:|
2841:b
2836:|
2828:,
2824:i
2818:x
2814:+
2810:|
2805:x
2797:a
2792:|
2784:i
2778:x
2774:+
2770:i
2761:)
2757:x
2749:a
2745:(
2742:=
2738:i
2732:a
2728:=
2720:|
2715:a
2710:|
2691:c
2687:b
2683:a
2678:O
2673:k
2669:j
2665:i
2659:x
2655:c
2651:b
2647:a
2626:X
2623:O
2614:,
2605:C
2602:O
2593:,
2584:B
2581:O
2572:,
2563:A
2560:O
2531:Δ
2526:0
2523:P
2518:Δ
2514:P
2500:)
2497:P
2494:(
2491:d
2485:)
2480:0
2476:P
2472:(
2469:d
2449:)
2442:P
2438:(
2435:d
2429:)
2424:0
2420:P
2416:(
2413:d
2393:)
2390:P
2387:(
2384:d
2378:)
2371:P
2367:(
2364:d
2334:P
2323:Δ
2319:P
2305:.
2302:)
2299:P
2296:(
2293:d
2286:|
2282:D
2279:A
2275:|
2271:=
2268:)
2263:0
2259:P
2255:(
2252:d
2238:Q
2234:P
2218:0
2214:P
2207:P
2187:.
2183:|
2179:D
2176:A
2172:|
2168:=
2163:0
2159:P
2155:(
2152:d
2141:D
2138:0
2135:Q
2132:0
2124:D
2119:0
2116:P
2110:0
2107:Q
2098:∠
2089:0
2086:Q
2081:B
2074:△
2067:△
2060:△
2055:0
2052:P
2046:0
2043:Q
2038:P
2033:0
2030:P
2016:0
2013:P
1990:|
1986:D
1983:Q
1979:|
1975:+
1971:|
1967:Q
1964:P
1960:|
1956:+
1952:|
1948:P
1945:A
1941:|
1937:=
1933:|
1929:C
1926:P
1922:|
1918:+
1914:|
1910:B
1907:P
1903:|
1899:+
1895:|
1891:A
1888:P
1884:|
1880:=
1877:)
1874:P
1871:(
1868:d
1855:C
1848:△
1841:△
1834:△
1830:Δ
1826:P
1819:△
1803:Δ
1799:A
1785:A
1779:P
1759:)
1756:P
1753:(
1750:d
1744:)
1741:A
1738:(
1735:d
1725:Δ
1721:P
1707:)
1704:P
1701:(
1698:d
1692:)
1689:A
1686:(
1683:d
1663:)
1656:P
1652:(
1649:d
1643:)
1640:A
1637:(
1634:d
1614:)
1611:P
1608:(
1605:d
1599:)
1592:P
1588:(
1585:d
1555:P
1544:Δ
1540:P
1526:.
1523:A
1517:P
1514:,
1505:P
1485:)
1482:P
1479:(
1476:d
1470:)
1467:A
1464:(
1461:d
1441:.
1438:)
1435:A
1432:(
1429:d
1426:=
1422:|
1418:F
1415:A
1411:|
1407:+
1403:|
1399:C
1396:A
1392:|
1378:△
1374:P
1355:|
1351:F
1348:Q
1344:|
1340:+
1336:|
1332:Q
1329:P
1325:|
1321:+
1317:|
1313:P
1310:C
1306:|
1302:=
1299:)
1296:P
1293:(
1290:d
1280:A
1273:△
1266:△
1259:△
1255:Q
1251:A
1247:Δ
1243:P
1236:△
1232:A
1211:.
1208:)
1205:P
1202:(
1199:d
1193:)
1186:P
1182:(
1179:d
1168:Ω
1160:Δ
1156:P
1128:)
1125:P
1122:(
1119:d
1113:)
1106:P
1102:(
1099:d
1089:A
1085:P
1071:)
1068:P
1065:(
1062:d
1056:)
1053:A
1050:(
1047:d
1044:=
1041:)
1034:P
1030:(
1027:d
1007:A
1004:=
997:P
986:C
982:B
978:P
974:P
970:Δ
966:Δ
962:P
945:.
941:|
937:C
934:B
930:|
926:+
922:|
918:C
915:A
911:|
907:+
903:|
899:B
896:A
892:|
884:|
880:B
877:X
873:|
869:+
865:|
861:X
858:C
854:|
850:+
846:|
842:C
839:A
835:|
831:+
827:|
823:B
820:A
816:|
812:=
808:|
804:B
801:X
797:|
793:+
789:|
785:X
782:A
778:|
774:+
770:|
766:B
763:A
759:|
740::
734:X
710:Ω
706:Δ
692:.
687:0
683:P
676:P
656:)
653:P
650:(
647:d
641:)
636:0
632:P
628:(
625:d
614:0
611:P
596:.
592:|
588:C
585:P
581:|
577:+
573:|
569:B
566:P
562:|
558:+
554:|
550:A
547:P
543:|
539:=
536:)
533:P
530:(
527:d
517:P
510:△
472:△
465:△
440:A
425:∠
405:∠
393:∠
389:F
385:A
378:△
374:A
366:∠
342:F
330:∠
318:∠
288:∠
280:∠
269:∠
257:∠
249:∠
214:∠
203:∠
199:A
184:△
177:F
162:△
147:△
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