4262:
3289:
2483:
1476:
1900:
4276:
2997:
2494:
2197:
1190:
2502:
1087:
1079:
242:
1539:
2186:
753:
496:
264:, which means that the optimal point is the only point where an infinitesimal movement towards one of the three reference points induces a reduction of the distance to that point that is equal to the sum of the induced changes in the distances to the two other points; in fact, in the Fermat problem, the advantage to reduce the distance from
3284:{\displaystyle {\begin{aligned}x&=\sin \angle f-{\frac {\overline {RA_{1}}}{\overline {RA_{2}}}}\times {\frac {\sin \angle d\sin(\angle e-\angle b)}{\sin \angle c}};\\y&={\frac {\overline {RA_{1}}}{\overline {RA_{2}}}}\times {\frac {\sin \angle d\cos(\angle e-\angle b)}{\sin \angle c}}-\cos \angle f;\end{aligned}}}
2478:{\displaystyle {\begin{aligned}\angle 1+\angle 2&=\angle C;\\\angle 3+\angle 4&=\angle A;\\\angle 5+\angle 6&=\angle B;\\\angle 1+\angle 6+\angle \alpha _{A}&=180^{\circ };\\\angle 2+\angle 3+\angle \alpha _{B}&=180^{\circ };\\\angle 4+\angle 5+\angle \alpha _{C}&=180^{\circ }.\end{aligned}}}
1471:{\displaystyle {\begin{aligned}\angle 1+\angle 2&=\angle C;\\\angle 3+\angle 4&=\angle A;\\\angle 5+\angle 6&=\angle B;\\\angle 1+\angle 6+\angle \alpha _{A}&=180^{\circ };\\\angle 2+\angle 3+\angle \alpha _{B}&=180^{\circ };\\\angle 4+\angle 5+\angle \alpha _{C}&=180^{\circ }.\end{aligned}}}
2002:
3907:
3481:
When the number of forces is larger than three, it is no longer possible to determine the angles separating the various forces without taking into account the geometry of the location polygon. Geometric and trigonometric methods are then powerless. Iterative optimizing methods are used in such cases.
1480:
Unfortunately, this system of six simultaneous equations with six unknowns is undetermined, and the possibility of the origins of the three vectors oriented towards the three attraction points not coinciding explains why. In the case of non-coincidence, we observe that all the six equations are still
3937:
In the case where everybody is attracted by a single attraction point (the rural market or the urban central business district), competition between the various bidders who all want to locate at the center will generate land values that will transform the unique attraction point of the system into a
151:
before 1640, and it can be seen as the true beginning of both location theory, and space-economy. Torricelli found a geometrical solution to this problem around 1645, but it still had no direct numerical solution more than 325 years later. E. Weiszfeld published a paper in 1937 with an algorithm for
172:
and each of the three other points is minimized. The Weber problem is a generalization of the Fermat problem since it involves both equal and unequal attractive forces (see below), while the Fermat problem only deals with equal attractive forces. It was first formulated, and solved geometrically in
152:
the Fermat-Weber problem. As the paper was published in Tohoku
Mathematical journal, and Weiszfeld immigrated to USA and changed his name to Vaszoni, his work was not widely known. Kuhn and Kuenne independently found a similar iterative solution for the general Fermat problem in 1962, and, in 1972,
4203:
Tellier, Luc-Normand, 2013, « Annexe 1: Solution géométrique du cas triangulaire du problème d’attraction–répulsion », annex of the paper of Pierre Hansen, Christophe Meyer and Luc-Normand
Tellier, « Modèles topodynamique et de la Nouvelle économie géographique : compatibilité,
1094:
More than 332 years separate the first formulation of the Fermat triangle problem and the discovery of its non-iterative numerical solution, while a geometrical solution existed for almost all that period of time. Is there an explanation for that? That explanation lies in the possibility of the
1069:
This solution is useless if one of the forces is greater than the sum of the two other ones or if the angles are not compatible. In some cases, no force is larger than the two other ones, and the angles are not compatible; then, the optimal location lies at the point that exerts the greater
4067:
Tellier, Luc-Normand, 2013, « Annexe 1 : Solution géométrique du cas triangulaire du problème d’attraction-répulsion », annex of the paper of Pierre Hansen, Christophe Meyer and Luc-Normand
Tellier, « Modèles topodynamique et de la Nouvelle économie géographique :
1895:{\displaystyle {\begin{aligned}\cos \angle \alpha _{A}=-{\frac {w_{B}^{2}+w_{C}^{2}-w_{A}^{2}}{2\,w_{B}w_{C}}};\\\cos \angle \alpha _{B}=-{\frac {w_{A}^{2}+w_{C}^{2}-w_{B}^{2}}{2\,w_{A}w_{C}}};\\\cos \angle \alpha _{C}=-{\frac {w_{A}^{2}+w_{B}^{2}-w_{C}^{2}}{2\,w_{A}w_{B}}};\end{aligned}}}
287:
According to an important theorem of
Euclidean geometry, in a convex quadrilateral inscribed in a circle, the opposite angles are supplementary (that is their sum is equal to 180°); that theorem can also take the following form: if we cut a circle with a chord
156:
found a direct numerical solution to the Fermat triangle problem, which is trigonometric. Kuhn and Kuenne's solution applies to the case of polygons having more than three sides, which is not the case with
Tellier's solution for reasons explained further on.
181:
in 1909. Kuhn and Kuenne's iterative solution found in 1962, and
Tellier's solution found in 1972 apply to the Weber triangle problem as well as to the Fermat one. Kuhn and Kuenne's solution applies also to the case of polygons having more than three sides.
1489:
In order to solve the problem, we must add to the six simultaneous equations a seventh requirement, which states that there should be no triangular hole in the middle of the location triangle. In other words, the origins of the three vectors must coincide.
3716:
3712:
of the input points are divided by the distances from each point to the approximation from the previous stage. As the unique optimal solution to a weighted least squares problem, each successive approximation may be found as a weighted average:
1485:
has disappeared because of the triangular hole that exists inside the triangle. In fact, as
Tellier (1972) has shown, that triangular hole had exactly the same proportions as the "forces triangles" we drew in Simpson's geometrical solution.
760:
A geometrical solution exists for the attraction-repulsion triangle problem. Its discovery is rather recent. That geometrical solution differs from the two previous ones since, in this case, the two constructed force triangles overlap the
50:, the geometric median of three points. For this reason it is sometimes called the Fermat–Weber problem, although the same name has also been used for the unweighted geometric median problem. The Weber problem is in turn generalized by the
2684:
2823:
3938:
repulsion point from the land value point of view, and, at the equilibrium, each inhabitant and activity will be located at the point where the attractive and the repulsive forces exerted by the center on them will cancel out.
2202:
1195:
2542:. Here as in the previous case, the possibility exists for the origins of the three vectors not to coincide. So the solution must require their coinciding. Tellier's trigonometric solution of this problem is the following:
3703:
2181:{\displaystyle {\begin{aligned}k&={\frac {\overline {CB}}{\overline {CA}}}\times {\frac {\sin \angle \alpha _{B}}{\sin \angle \alpha _{A}}},\\k'&=(\angle A+\angle B+\angle \alpha _{C})-180^{\circ }.\end{aligned}}}
1997:
3424:
3958:
in 2008. The concept of attractive force is akin to the NEG concept of agglomeration or centripetal force, and the concept of repulsive force is akin to the NEG concept of dispersal or centrifugal force.
3002:
2007:
3352:
507:
in 1750) directly derives from
Torricelli's solution. Simpson and Weber stressed the fact that, in a total transportation minimization problem, the advantage to get closer to each attraction point
1544:
3570:
229:
cancel each other out as it must do at the optimum. It constitutes a generalization of both the Fermat and Weber problems. It was first formulated and solved, in the triangle case, in 1985 by
2880:
2992:
3471:
2933:
51:
3494:. Their method is valid for the Fermat and Weber problems involving many forces, but not for the attraction–repulsion problem. In this method, to find an approximation to the point
482:
triangle, as well as two other circles round these equilateral triangles, and to determine the location where the three circles intersect; at that location, the angles between the
2556:
2698:
3596:
4182:
Tellier, Luc-Normand and Boris
Polanski, 1989, "The Weber Problem: Frequency of Different Solution Types and Extension to Repulsive Forces and Dynamic Processes",
3902:{\displaystyle y_{j+1}={\frac {\displaystyle \sum _{i=1}^{n}{\frac {w_{i}x_{i}}{|x_{i}-y_{j}|}}}{\displaystyle \sum _{i=1}^{n}{\frac {w_{i}}{|x_{i}-y_{j}|}}}}}
2513:
Tellier (1985) extended the Fermat–Weber problem to the case of repulsive forces. Let us examine the triangle case where there are two attractive forces
4165:
Kuhn, Harold W. and Robert E. Kuenne, 1962, "An
Efficient Algorithm for the Numerical Solution of the Generalized Weber Problem in Spatial Economics."
4107:
Kuhn, Harold W. and Robert E. Kuenne, 1962, "An Efficient Algorithm for the Numerical Solution of the Generalized Weber Problem in Spatial Economics."
3998:
Kuhn, Harold W. and Robert E. Kuenne, 1962, "An Efficient Algorithm for the Numerical Solution of the Generalized Weber Problem in Spatial Economics."
3363:
1095:
origins of the three vectors oriented towards the three attraction points not coinciding. If those origins do coincide and lie at the optimum location
1919:
4307:
3300:
740:
side, and a third circumference can be traced round that triangle. That third circumference crosses the two previous ones at the same point
260:
is at its optimal location when any significant move out of that location induces a net increase of the total distance to reference points
3501:
233:. In 1992, Chen, Hansen, Jaumard and Tuy found a solution to the Tellier problem for the case of polygons having more than three sides.
3955:
3950:. It is seen by Ottaviano and Thisse (2005) as a prelude to the New Economic Geography (NEG) that developed in the 1990s, and earned
2837:
2952:
3918:
For the attraction–repulsion problem one has instead to resort to the algorithm proposed by Chen, Hansen, Jaumard and Tuy (1992).
3435:
2894:
46:, which assumes transportation costs per unit distance are the same for all destination points, and the problem of computing the
556:
were equilateral because the three attractive forces were equal, in the Weber triangle problem case, the constructed triangles
3483:
515:
depends on what is carried and on its transportation cost. Consequently, the advantage of getting one kilometer closer to
3930:, repulsive forces are omnipresent. Land values are the main illustration of them. In fact a substantial portion of
357:
straight lines must be equal to 360° / 3 = 120°. Torricelli deduced from that conclusion that:
4172:
Ottaviano, Gianmarco and Jacques-François Thisse, 2005, « New Economic Geography: what about the N? »,
4137:
Ottaviano, Gianmarco and Jacques-François Thisse, 2005, « New Economic Geography: what about the N? »,
35:. It requires finding a point in the plane that minimizes the sum of the transportation costs from this point to
3977:
3947:
541:
Simpson demonstrated that, in the same way as, in the Fermat triangle problem case, the constructed triangles
39:
destination points, where different destination points are associated with different costs per unit distance.
4281:
4250:
4297:
3921:
4302:
4245:
1181:, whose values depend only on the relative magnitude of the three attractive forces pointing towards the
503:
Simpson's geometrical solution of the so-called "Weber triangle problem" (which was first formulated by
4317:
1536:
cancel each other to ensure equilibrium. This is done by means of the following independent equations:
147:
and each of the three other points is minimized. It was formulated by the famous French mathematician
54:, which allows some of the costs to be negative, so that greater distance from some points is better.
4312:
3581:
is found, and then at each stage of the algorithm is moved closer to the optimal solution by setting
4240:
3487:
353:
It can be proved that the first observation implies that, at the optimum, the angles between the
57:
2488:
249:
747:
3476:
1073:
4068:
compatibilité, convergence et avantages comparés », in Marc-Urbain Proulx (ed.), 2013,
2679:{\displaystyle \cos \angle e=-{\frac {w_{A1}^{2}+w_{A2}^{2}-w_{R}^{2}}{2\,w_{A1}w_{A2}}};}
486:
straight lines is necessarily equal to 120°, which proves that it is the optimal location.
8:
2818:{\displaystyle \cos \angle p=-{\frac {w_{A1}^{2}+w_{R}^{2}-w_{A2}^{2}}{2\,w_{A1}w_{R}}};}
1138:
angles. It is easy to write the following six equations linking six unknowns (the angles
230:
153:
4267:
3922:
Interpretation of the land rent theory in the light of the attraction–repulsion problem
185:
In its simplest version, the attraction-repulsion problem consists in locating a point
4261:
252:’s geometrical solution of the Fermat triangle problem stems from two observations:
4155:
4121:
3491:
1493:
Tellier's solution of the Fermat and Weber triangle problems involves three steps:
236:
148:
43:
3941:
3931:
3927:
32:
3978:"Sur le point pour lequel la Somme des distances de n points donnés est minimum"
582:
triangle, must be proportional to the attractive forces of the location system.
414:
triangle (because each angle of an equilateral triangle is equal to 60°), where
3912:
504:
490:
174:
4189:
Tellier, Luc-Normand, 1972, "The Weber Problem: Solution and Interpretation",
4081:
Tellier, Luc-Normand, 1972, "The Weber Problem: Solution and Interpretation",
4011:
Tellier, Luc-Normand, 1972, "The Weber Problem: Solution and Interpretation",
3698:{\displaystyle \sum _{i=1}^{n}{\frac {w_{i}}{\|x_{i}-y_{j}\|}}\|x_{i}-y\|^{2}}
58:
Definition and history of the Fermat, Weber, and attraction-repulsion problems
4291:
4204:
convergence et avantages comparés », in Marc-Urbain Proulx (ed.), 2013,
2489:
Tellier’s trigonometric solution of the triangle attraction-repulsion problem
756:
Tellier's geometrical solution of the attraction-repulsion triangle problem.
3951:
748:
Tellier’s geometrical solution of the attraction-repulsion triangle problem
178:
47:
28:
3477:
Iterative solutions of the Fermat, Weber and attraction-repulsion problems
1074:
Tellier’s trigonometric solution of the Fermat and Weber triangle problems
4222:
Wesolowski, Georges, 1993, «The Weber problem: History and perspective»,
1039:
is located at the intersection of the two circumferences drawn round the
703:
is located at the intersection of the two circumferences drawn round the
4179:
Simpson, Thomas, 1750, The Doctrine and Application of Fluxions, London.
393:
triangle must be equal to (180° − 120°) = 60°;
268:
by one kilometer is equal to the advantage to reduce the distance from
4158:
and Hoang Tuy, 1992, "Weber's Problem with Attraction and Repulsion,"
4124:
and Hoang Tuy, 1992, "Weber's Problem with Attraction and Repulsion,"
160:
The Weber problem consists, in the triangle case, in locating a point
135:
In the triangle case, the Fermat problem consists in locating a point
802:, a repulsion one), while, in the preceding cases, they never did.
20:
276:
by the same length; in other words, the activity to be located at
3593:
to be the point minimizing the sum of weighted squared distances
3419:{\displaystyle \angle 5=180^{\circ }-\angle b-\angle c-\angle 1;}
245:
Torricelli's geometrical solution of the Fermat triangle problem.
237:
Torricelli’s geometrical solution of the Fermat triangle problem
4275:
3942:
The attraction–repulsion problem and the New Economic Geography
3934:, both rural and urban, can be summed up in the following way.
429:
points of the circumference of that circle that lie within the
168:
in such a way that the sum of the transportation costs between
425:
triangle, and draw a circle round that triangle; then all the
1992:{\displaystyle \tan \angle 3={\frac {k\sin k'}{1+k\cos k'}};}
499:
Simpson's geometrical solution of the Weber triangle problem.
4198:Économie spatiale: rationalité économique de l'espace habité
4096:Économie spatiale: rationalité économique de l'espace habité
4056:Économie spatiale: rationalité économique de l'espace habité
1090:
The case of non-coincidence of the vertices of the α angles.
491:
Simpson’s geometrical solution of the Weber triangle problem
2190:
Solve the following system of simultaneous equations where
209:
in such a way that the attractive forces exerted by points
446:
The same reasoning can be made with respect to triangles
3915:
provides an experimental solution of the Weber problem.
3347:{\displaystyle \angle 1=180^{\circ }-\angle e-\angle 3;}
2497:
The angles of the attraction-repulsion triangle problem.
241:
3482:
Kuhn and Kuenne (1962) suggested an algorithm based on
2941:(this equation derives from the requirement that point
2501:
2493:
1908:(this equation derives from the requirement that point
1086:
1078:
752:
495:
4219:, Chicago, Chicago University Press, 1929, 256 pages.
4045:, Chicago, Chicago University Press, 1929, 256 pages.
3825:
3741:
3719:
3599:
3504:
3438:
3366:
3303:
3000:
2955:
2897:
2840:
2701:
2559:
2200:
2005:
1922:
1542:
1193:
4257:
143:
in such a way that the sum of the distances between
460:This leads to draw two other equilateral triangles
3946:The Tellier problem preceded the emergence of the
3901:
3697:
3565:{\displaystyle \sum _{i=1}^{n}w_{i}\,\|x_{i}-y\|,}
3564:
3465:
3418:
3346:
3283:
2986:
2927:
2874:
2817:
2678:
2477:
2180:
1991:
1894:
1470:
106:Direct numerical solution of the triangle problem
407:angle is equal to 120° is to draw an equilateral
4289:
4215:, Tübingen, J.C.B. Mohr) — English translation:
4041:, Tübingen, J.C.B. Mohr) — English translation:
379:convex quadrilateral inscribed in a circle, the
2875:{\displaystyle \angle c=180^{\circ }-\angle p;}
1521:that are such that the three attractive forces
4206:Sciences du territoire II : méthodologies
4200:, Chicoutimi, Gaëtan Morin éditeur, 280 pages.
4098:, Chicoutimi, Gaëtan Morin éditeur, 280 pages.
4070:Sciences du territoire II : méthodologies
4058:, Chicoutimi, Gaëtan Morin éditeur, 280 pages.
2987:{\displaystyle \tan \angle 3={\frac {x}{y}};}
988:side is proportional to the attractive force
963:side is proportional to the attractive force
875:side is proportional to the attractive force
850:side is proportional to the attractive force
685:side is proportional to the attractive force
670:side is proportional to the attractive force
655:side is proportional to the attractive force
630:side is proportional to the attractive force
615:side is proportional to the attractive force
600:side is proportional to the attractive force
396:One way to determine the set of locations of
92:Geometrical solution of the triangle problem
4208:, Québec, Presses de l’Université du Québec.
4072:, Québec, Presses de l’Université du Québec.
3686:
3666:
3660:
3634:
3556:
3537:
1021:side is proportional to the repulsive force
908:side is proportional to the repulsive force
126:E. Weiszfeld (1937), Kuhn and Kuenne (1962)
123:E. Weiszfeld (1937), Kuhn and Kuenne (1962)
120:Iterative numerical solution of the problem
16:Problem of minimizing sum of transport costs
3466:{\displaystyle \angle 2=\angle a-\angle 5.}
2928:{\displaystyle \angle d=\angle e-\angle c;}
334:angles are also equal for any chosen point
225:, and the repulsive force exerted by point
3975:
3956:Nobel Memorial Prize in Economic Sciences
3572:an initial approximation to the solution
3536:
3498:minimizing the weighted sum of distances
2785:
2643:
1861:
1746:
1631:
538:angles no more need to be equal to 120°.
4217:The Theory of the Location of Industries
4043:The Theory of the Location of Industries
4026:The Doctrine and Application of Fluxions
2500:
2492:
1085:
1077:
751:
494:
240:
31:, is one of the most famous problems in
317:angle is the same for any chosen point
4290:
2505:The case of non-coincidence of points
1110:location triangle form the six angles
272:by one kilometer or the distance from
4308:Mathematical optimization in business
1481:valid. However, the optimal location
1157:, whose values are given, and angles
375:angle is equal to 120°, generates an
294:, we get two circle arcs, let us say
177:in 1750. It was later popularized by
129:Chen, Hansen, Jaumard and Tuy (1992)
3484:iteratively reweighted least squares
736:triangle, can be drawn based on the
13:
4193:, vol. 4, no. 3, pp. 215–233.
3457:
3448:
3439:
3407:
3398:
3389:
3367:
3335:
3326:
3304:
3268:
3250:
3233:
3224:
3209:
3127:
3110:
3101:
3086:
3021:
2962:
2916:
2907:
2898:
2863:
2841:
2708:
2566:
2438:
2429:
2420:
2383:
2374:
2365:
2328:
2319:
2310:
2297:
2284:
2275:
2262:
2249:
2240:
2227:
2214:
2205:
2142:
2133:
2124:
2083:
2062:
1929:
1783:
1668:
1553:
1431:
1422:
1413:
1376:
1367:
1358:
1321:
1312:
1303:
1290:
1277:
1268:
1255:
1242:
1233:
1220:
1207:
1198:
42:The Weber problem generalizes the
14:
4329:
4233:
4186:, vol 29, no. 3, p. 387–405.
1114:, and the three vectors form the
73:The attraction-repulsion problem
4274:
4260:
4213:Über den Standort der Industrien
4154:Chen, Pey-Chun, Hansen, Pierre,
4120:Chen, Pey-Chun, Hansen, Pierre,
4039:Über den Standort der Industrien
1142:) with six known values (angles
1082:The angles of the Weber problem.
4131:
4114:
4101:
1099:, the vectors oriented towards
61:
4088:
4075:
4061:
4048:
4031:
4018:
4005:
3992:
3969:
3889:
3861:
3817:
3789:
3239:
3221:
3116:
3098:
2155:
2121:
1:
4282:Business and economics portal
4148:
4085:, vol. 4, no. 3, pp. 215–233.
4015:, vol. 4, no. 3, pp. 215–233.
2937:Determine the value of angle
1904:Determine the value of angle
189:with respect to three points
164:with respect to three points
139:with respect to three points
4196:Tellier, Luc-Normand, 1985,
4094:Tellier, Luc-Normand, 1985,
4054:Tellier, Luc-Normand, 1985,
3191:
3171:
3068:
3048:
2044:
2031:
935:, which partly overlaps the
922:In the constructed triangle
822:, which partly overlaps the
809:In the constructed triangle
805:This solution is such that:
644:In the constructed triangle
589:In the constructed triangle
52:attraction–repulsion problem
7:
4246:Encyclopedia of Mathematics
4184:Journal of Regional Science
4167:Journal of Regional Science
4160:Journal of Regional Science
4126:Journal of Regional Science
4109:Journal of Regional Science
4000:Journal of Regional Science
3982:Tohoku Mathematical Journal
798:are attraction points, and
718:A third triangle of forces
585:The solution is such that:
10:
4334:
4174:Environment and Planning A
4139:Environment and Planning A
3705:where the initial weights
2535:, and one repulsive force
128:
125:
122:
119:
114:
111:
108:
105:
100:
97:
94:
91:
86:
83:
80:
77:
2945:must coincide with point
1912:must coincide with point
780:location triangle (where
436:circle are such that the
349:angles are supplementary.
72:
69:
66:
64:
3962:
1028:pushing away from point
915:pushing away from point
575:are located outside the
475:are located outside the
280:is equally attracted by
4226:, Vol. 1, p. 5–23.
4024:Simpson, Thomas, 1750,
1103:, and the sides of the
954:location triangle, the
841:location triangle, the
729:is located outside the
443:angle is equal to 120°;
418:is located outside the
3976:Weiszfeld, E. (1937).
3948:New Economic Geography
3903:
3846:
3762:
3699:
3620:
3566:
3525:
3467:
3420:
3348:
3285:
2988:
2929:
2876:
2819:
2680:
2510:
2498:
2479:
2182:
1993:
1896:
1472:
1140:∠1, ∠2, ∠3, ∠4, ∠5, ∠6
1112:∠1, ∠2, ∠3, ∠4, ∠5, ∠6
1091:
1083:
1065:constructed triangles.
757:
714:constructed triangles.
500:
250:Evangelista Torricelli
246:
173:the triangle case, by
4211:Weber, Alfred, 1909,
4191:Geographical Analysis
4083:Geographical Analysis
4037:Weber, Alfred, 1909,
4013:Geographical Analysis
3904:
3826:
3742:
3700:
3600:
3567:
3505:
3488:Weiszfeld's algorithm
3468:
3421:
3349:
3286:
2989:
2930:
2877:
2820:
2681:
2504:
2496:
2480:
2183:
1994:
1897:
1497:Determine the angles
1473:
1089:
1081:
755:
498:
244:
81:Fermat (before 1640)
3717:
3597:
3502:
3436:
3364:
3301:
2998:
2953:
2895:
2838:
2699:
2557:
2198:
2003:
1920:
1540:
1191:
1185:attraction points):
78:First formulated by
4298:Applied mathematics
2779:
2758:
2740:
2637:
2619:
2598:
1855:
1837:
1819:
1740:
1722:
1704:
1625:
1607:
1589:
231:Luc-Normand Tellier
67:The Fermat problem
4303:Economic geography
4268:Mathematics portal
3899:
3896:
3824:
3695:
3562:
3492:unweighted problem
3463:
3416:
3344:
3281:
3279:
2984:
2925:
2872:
2815:
2762:
2744:
2723:
2676:
2623:
2602:
2581:
2511:
2499:
2475:
2473:
2178:
2176:
1989:
1892:
1890:
1841:
1823:
1805:
1726:
1708:
1690:
1611:
1593:
1575:
1468:
1466:
1092:
1084:
1070:attractive force.
1035:The optimal point
758:
699:The optimal point
501:
247:
95:Torricelli (1645)
70:The Weber problem
4318:Facility location
4156:Jaumard, Brigitte
4122:Jaumard, Brigitte
3932:land value theory
3928:spatial economics
3897:
3894:
3822:
3664:
3257:
3195:
3194:
3174:
3134:
3072:
3071:
3051:
2979:
2810:
2671:
2097:
2048:
2047:
2034:
1984:
1883:
1768:
1653:
1000:pointing towards
975:pointing towards
887:pointing towards
862:pointing towards
692:pointing towards
677:pointing towards
662:pointing towards
637:pointing towards
622:pointing towards
607:pointing towards
133:
132:
4325:
4313:Regional science
4284:
4279:
4278:
4270:
4265:
4264:
4254:
4224:Location Science
4142:
4135:
4129:
4118:
4112:
4105:
4099:
4092:
4086:
4079:
4073:
4065:
4059:
4052:
4046:
4035:
4029:
4022:
4016:
4009:
4003:
3996:
3990:
3989:
3984:. First Series.
3973:
3926:In the world of
3908:
3906:
3905:
3900:
3898:
3895:
3893:
3892:
3887:
3886:
3874:
3873:
3864:
3858:
3857:
3848:
3845:
3840:
3823:
3821:
3820:
3815:
3814:
3802:
3801:
3792:
3786:
3785:
3784:
3775:
3774:
3764:
3761:
3756:
3740:
3735:
3734:
3711:
3704:
3702:
3701:
3696:
3694:
3693:
3678:
3677:
3665:
3663:
3659:
3658:
3646:
3645:
3632:
3631:
3622:
3619:
3614:
3592:
3580:
3571:
3569:
3568:
3563:
3549:
3548:
3535:
3534:
3524:
3519:
3497:
3472:
3470:
3469:
3464:
3431:
3425:
3423:
3422:
3417:
3385:
3384:
3359:
3353:
3351:
3350:
3345:
3322:
3321:
3296:
3290:
3288:
3287:
3282:
3280:
3258:
3256:
3242:
3201:
3196:
3190:
3189:
3188:
3175:
3170:
3169:
3168:
3155:
3154:
3135:
3133:
3119:
3078:
3073:
3067:
3066:
3065:
3052:
3047:
3046:
3045:
3032:
3031:
2993:
2991:
2990:
2985:
2980:
2972:
2948:
2944:
2940:
2934:
2932:
2931:
2926:
2890:
2884:Determine angle
2881:
2879:
2878:
2873:
2859:
2858:
2833:
2827:Determine angle
2824:
2822:
2821:
2816:
2811:
2809:
2808:
2807:
2798:
2797:
2780:
2778:
2773:
2757:
2752:
2739:
2734:
2721:
2694:
2688:Determine angle
2685:
2683:
2682:
2677:
2672:
2670:
2669:
2668:
2656:
2655:
2638:
2636:
2631:
2618:
2613:
2597:
2592:
2579:
2552:
2546:Determine angle
2541:
2534:
2508:
2484:
2482:
2481:
2476:
2474:
2467:
2466:
2450:
2449:
2412:
2411:
2395:
2394:
2357:
2356:
2340:
2339:
2193:
2187:
2185:
2184:
2179:
2177:
2170:
2169:
2154:
2153:
2113:
2098:
2096:
2095:
2094:
2075:
2074:
2073:
2054:
2049:
2043:
2035:
2030:
2022:
2021:
1998:
1996:
1995:
1990:
1985:
1983:
1982:
1958:
1957:
1939:
1915:
1911:
1907:
1901:
1899:
1898:
1893:
1891:
1884:
1882:
1881:
1880:
1871:
1870:
1856:
1854:
1849:
1836:
1831:
1818:
1813:
1803:
1795:
1794:
1769:
1767:
1766:
1765:
1756:
1755:
1741:
1739:
1734:
1721:
1716:
1703:
1698:
1688:
1680:
1679:
1654:
1652:
1651:
1650:
1641:
1640:
1626:
1624:
1619:
1606:
1601:
1588:
1583:
1573:
1565:
1564:
1535:
1520:
1484:
1477:
1475:
1474:
1469:
1467:
1460:
1459:
1443:
1442:
1405:
1404:
1388:
1387:
1350:
1349:
1333:
1332:
1184:
1180:
1156:
1141:
1137:
1113:
1109:
1102:
1098:
1064:
1051:
1038:
1031:
1027:
1020:
1008:
999:
987:
983:
974:
962:
953:
934:
918:
914:
907:
895:
886:
874:
870:
861:
849:
840:
821:
801:
797:
788:
779:
743:
739:
735:
728:
724:
713:
702:
695:
691:
684:
680:
676:
669:
665:
661:
654:
650:
640:
636:
629:
625:
621:
614:
610:
606:
599:
595:
581:
574:
570:
555:
537:
523:varies, and the
522:
518:
514:
510:
485:
481:
474:
470:
456:
442:
435:
428:
424:
417:
413:
406:
399:
392:
385:
378:
374:
367:
361:If any triangle
356:
348:
338:; moreover, the
337:
333:
326:
325:
320:
316:
309:
308:
303:
302:
298:
293:
292:
283:
279:
275:
271:
267:
263:
259:
228:
224:
208:
204:
188:
171:
167:
163:
149:Pierre de Fermat
146:
142:
138:
62:
44:geometric median
38:
4333:
4332:
4328:
4327:
4326:
4324:
4323:
4322:
4288:
4287:
4280:
4273:
4266:
4259:
4241:"Weber problem"
4239:
4236:
4230:
4151:
4146:
4145:
4136:
4132:
4119:
4115:
4106:
4102:
4093:
4089:
4080:
4076:
4066:
4062:
4053:
4049:
4036:
4032:
4023:
4019:
4010:
4006:
3997:
3993:
3974:
3970:
3965:
3944:
3924:
3888:
3882:
3878:
3869:
3865:
3860:
3859:
3853:
3849:
3847:
3841:
3830:
3816:
3810:
3806:
3797:
3793:
3788:
3787:
3780:
3776:
3770:
3766:
3765:
3763:
3757:
3746:
3739:
3724:
3720:
3718:
3715:
3714:
3710:
3706:
3689:
3685:
3673:
3669:
3654:
3650:
3641:
3637:
3633:
3627:
3623:
3621:
3615:
3604:
3598:
3595:
3594:
3591:
3582:
3579:
3573:
3544:
3540:
3530:
3526:
3520:
3509:
3503:
3500:
3499:
3495:
3479:
3437:
3434:
3433:
3429:
3380:
3376:
3365:
3362:
3361:
3357:
3317:
3313:
3302:
3299:
3298:
3294:
3278:
3277:
3243:
3202:
3200:
3184:
3180:
3176:
3164:
3160:
3156:
3153:
3146:
3140:
3139:
3120:
3079:
3077:
3061:
3057:
3053:
3041:
3037:
3033:
3030:
3008:
3001:
2999:
2996:
2995:
2971:
2954:
2951:
2950:
2946:
2942:
2938:
2896:
2893:
2892:
2885:
2854:
2850:
2839:
2836:
2835:
2828:
2803:
2799:
2790:
2786:
2781:
2774:
2766:
2753:
2748:
2735:
2727:
2722:
2720:
2700:
2697:
2696:
2689:
2661:
2657:
2648:
2644:
2639:
2632:
2627:
2614:
2606:
2593:
2585:
2580:
2578:
2558:
2555:
2554:
2547:
2540:
2536:
2533:
2523:
2514:
2506:
2491:
2472:
2471:
2462:
2458:
2451:
2445:
2441:
2417:
2416:
2407:
2403:
2396:
2390:
2386:
2362:
2361:
2352:
2348:
2341:
2335:
2331:
2307:
2306:
2290:
2272:
2271:
2255:
2237:
2236:
2220:
2201:
2199:
2196:
2195:
2191:
2175:
2174:
2165:
2161:
2149:
2145:
2114:
2106:
2103:
2102:
2090:
2086:
2076:
2069:
2065:
2055:
2053:
2036:
2023:
2020:
2013:
2006:
2004:
2001:
2000:
1975:
1959:
1950:
1940:
1938:
1921:
1918:
1917:
1913:
1909:
1905:
1889:
1888:
1876:
1872:
1866:
1862:
1857:
1850:
1845:
1832:
1827:
1814:
1809:
1804:
1802:
1790:
1786:
1774:
1773:
1761:
1757:
1751:
1747:
1742:
1735:
1730:
1717:
1712:
1699:
1694:
1689:
1687:
1675:
1671:
1659:
1658:
1646:
1642:
1636:
1632:
1627:
1620:
1615:
1602:
1597:
1584:
1579:
1574:
1572:
1560:
1556:
1543:
1541:
1538:
1537:
1534:
1530:
1526:
1522:
1518:
1511:
1504:
1498:
1482:
1465:
1464:
1455:
1451:
1444:
1438:
1434:
1410:
1409:
1400:
1396:
1389:
1383:
1379:
1355:
1354:
1345:
1341:
1334:
1328:
1324:
1300:
1299:
1283:
1265:
1264:
1248:
1230:
1229:
1213:
1194:
1192:
1189:
1188:
1182:
1178:
1171:
1164:
1158:
1143:
1139:
1135:
1128:
1121:
1115:
1111:
1104:
1100:
1096:
1076:
1060:
1053:
1047:
1040:
1036:
1029:
1026:
1022:
1016:
1010:
1007:
1001:
998:
989:
985:
982:
976:
973:
964:
961:
955:
949:
943:
936:
930:
923:
916:
913:
909:
903:
897:
894:
888:
885:
876:
872:
869:
863:
860:
851:
848:
842:
836:
830:
823:
817:
810:
799:
796:
790:
787:
781:
775:
769:
762:
750:
741:
737:
730:
726:
719:
704:
700:
693:
690:
686:
682:
678:
675:
671:
667:
663:
660:
656:
652:
645:
638:
635:
631:
627:
623:
620:
616:
612:
608:
605:
601:
597:
590:
576:
572:
557:
542:
524:
520:
516:
512:
508:
493:
483:
476:
472:
461:
447:
437:
430:
426:
419:
415:
408:
401:
397:
387:
380:
376:
369:
362:
354:
339:
335:
328:
323:
322:
318:
311:
306:
305:
300:
296:
295:
290:
289:
281:
277:
273:
269:
265:
261:
257:
239:
226:
223:
216:
210:
206:
203:
196:
190:
186:
169:
165:
161:
144:
140:
136:
115:Tellier (1985)
112:Tellier (1972)
109:Tellier (1972)
101:Tellier (2013)
98:Simpson (1750)
87:Tellier (1985)
84:Simpson (1750)
60:
36:
33:location theory
17:
12:
11:
5:
4331:
4321:
4320:
4315:
4310:
4305:
4300:
4286:
4285:
4271:
4256:
4255:
4235:
4234:External links
4232:
4228:
4227:
4220:
4209:
4201:
4194:
4187:
4180:
4177:
4176:37, 1707–1725.
4170:
4163:
4150:
4147:
4144:
4143:
4141:37, 1707–1725.
4130:
4113:
4100:
4087:
4074:
4060:
4047:
4030:
4017:
4004:
3991:
3967:
3966:
3964:
3961:
3943:
3940:
3923:
3920:
3913:Varignon frame
3891:
3885:
3881:
3877:
3872:
3868:
3863:
3856:
3852:
3844:
3839:
3836:
3833:
3829:
3819:
3813:
3809:
3805:
3800:
3796:
3791:
3783:
3779:
3773:
3769:
3760:
3755:
3752:
3749:
3745:
3738:
3733:
3730:
3727:
3723:
3708:
3692:
3688:
3684:
3681:
3676:
3672:
3668:
3662:
3657:
3653:
3649:
3644:
3640:
3636:
3630:
3626:
3618:
3613:
3610:
3607:
3603:
3586:
3577:
3561:
3558:
3555:
3552:
3547:
3543:
3539:
3533:
3529:
3523:
3518:
3515:
3512:
3508:
3478:
3475:
3474:
3473:
3462:
3459:
3456:
3453:
3450:
3447:
3444:
3441:
3426:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3383:
3379:
3375:
3372:
3369:
3354:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3320:
3316:
3312:
3309:
3306:
3291:
3276:
3273:
3270:
3267:
3264:
3261:
3255:
3252:
3249:
3246:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3199:
3193:
3187:
3183:
3179:
3173:
3167:
3163:
3159:
3152:
3149:
3147:
3145:
3142:
3141:
3138:
3132:
3129:
3126:
3123:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3076:
3070:
3064:
3060:
3056:
3050:
3044:
3040:
3036:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3009:
3007:
3004:
3003:
2983:
2978:
2975:
2970:
2967:
2964:
2961:
2958:
2935:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2882:
2871:
2868:
2865:
2862:
2857:
2853:
2849:
2846:
2843:
2825:
2814:
2806:
2802:
2796:
2793:
2789:
2784:
2777:
2772:
2769:
2765:
2761:
2756:
2751:
2747:
2743:
2738:
2733:
2730:
2726:
2719:
2716:
2713:
2710:
2707:
2704:
2686:
2675:
2667:
2664:
2660:
2654:
2651:
2647:
2642:
2635:
2630:
2626:
2622:
2617:
2612:
2609:
2605:
2601:
2596:
2591:
2588:
2584:
2577:
2574:
2571:
2568:
2565:
2562:
2538:
2528:
2518:
2490:
2487:
2486:
2485:
2470:
2465:
2461:
2457:
2454:
2452:
2448:
2444:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2418:
2415:
2410:
2406:
2402:
2399:
2397:
2393:
2389:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2363:
2360:
2355:
2351:
2347:
2344:
2342:
2338:
2334:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2308:
2305:
2302:
2299:
2296:
2293:
2291:
2289:
2286:
2283:
2280:
2277:
2274:
2273:
2270:
2267:
2264:
2261:
2258:
2256:
2254:
2251:
2248:
2245:
2242:
2239:
2238:
2235:
2232:
2229:
2226:
2223:
2221:
2219:
2216:
2213:
2210:
2207:
2204:
2203:
2188:
2173:
2168:
2164:
2160:
2157:
2152:
2148:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2115:
2112:
2109:
2105:
2104:
2101:
2093:
2089:
2085:
2082:
2079:
2072:
2068:
2064:
2061:
2058:
2052:
2046:
2042:
2039:
2033:
2029:
2026:
2019:
2016:
2014:
2012:
2009:
2008:
1988:
1981:
1978:
1974:
1971:
1968:
1965:
1962:
1956:
1953:
1949:
1946:
1943:
1937:
1934:
1931:
1928:
1925:
1902:
1887:
1879:
1875:
1869:
1865:
1860:
1853:
1848:
1844:
1840:
1835:
1830:
1826:
1822:
1817:
1812:
1808:
1801:
1798:
1793:
1789:
1785:
1782:
1779:
1776:
1775:
1772:
1764:
1760:
1754:
1750:
1745:
1738:
1733:
1729:
1725:
1720:
1715:
1711:
1707:
1702:
1697:
1693:
1686:
1683:
1678:
1674:
1670:
1667:
1664:
1661:
1660:
1657:
1649:
1645:
1639:
1635:
1630:
1623:
1618:
1614:
1610:
1605:
1600:
1596:
1592:
1587:
1582:
1578:
1571:
1568:
1563:
1559:
1555:
1552:
1549:
1546:
1545:
1532:
1528:
1524:
1516:
1509:
1502:
1463:
1458:
1454:
1450:
1447:
1445:
1441:
1437:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1411:
1408:
1403:
1399:
1395:
1392:
1390:
1386:
1382:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1356:
1353:
1348:
1344:
1340:
1337:
1335:
1331:
1327:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1301:
1298:
1295:
1292:
1289:
1286:
1284:
1282:
1279:
1276:
1273:
1270:
1267:
1266:
1263:
1260:
1257:
1254:
1251:
1249:
1247:
1244:
1241:
1238:
1235:
1232:
1231:
1228:
1225:
1222:
1219:
1216:
1214:
1212:
1209:
1206:
1203:
1200:
1197:
1196:
1176:
1169:
1162:
1133:
1126:
1119:
1075:
1072:
1067:
1066:
1058:
1045:
1033:
1024:
1014:
1005:
993:
980:
968:
959:
947:
941:
928:
920:
911:
901:
892:
880:
867:
855:
846:
834:
828:
815:
794:
785:
773:
767:
749:
746:
716:
715:
697:
688:
673:
658:
642:
633:
618:
603:
505:Thomas Simpson
492:
489:
488:
487:
458:
444:
400:for which the
394:
351:
350:
321:, and, on arc
285:
238:
235:
221:
214:
201:
194:
175:Thomas Simpson
131:
130:
127:
124:
121:
117:
116:
113:
110:
107:
103:
102:
99:
96:
93:
89:
88:
85:
82:
79:
75:
74:
71:
68:
65:
59:
56:
27:, named after
15:
9:
6:
4:
3:
2:
4330:
4319:
4316:
4314:
4311:
4309:
4306:
4304:
4301:
4299:
4296:
4295:
4293:
4283:
4277:
4272:
4269:
4263:
4258:
4252:
4248:
4247:
4242:
4238:
4237:
4231:
4225:
4221:
4218:
4214:
4210:
4207:
4202:
4199:
4195:
4192:
4188:
4185:
4181:
4178:
4175:
4171:
4168:
4164:
4161:
4157:
4153:
4152:
4140:
4134:
4127:
4123:
4117:
4110:
4104:
4097:
4091:
4084:
4078:
4071:
4064:
4057:
4051:
4044:
4040:
4034:
4027:
4021:
4014:
4008:
4001:
3995:
3987:
3983:
3979:
3972:
3968:
3960:
3957:
3953:
3949:
3939:
3935:
3933:
3929:
3919:
3916:
3914:
3909:
3883:
3879:
3875:
3870:
3866:
3854:
3850:
3842:
3837:
3834:
3831:
3827:
3811:
3807:
3803:
3798:
3794:
3781:
3777:
3771:
3767:
3758:
3753:
3750:
3747:
3743:
3736:
3731:
3728:
3725:
3721:
3690:
3682:
3679:
3674:
3670:
3655:
3651:
3647:
3642:
3638:
3628:
3624:
3616:
3611:
3608:
3605:
3601:
3589:
3585:
3576:
3559:
3553:
3550:
3545:
3541:
3531:
3527:
3521:
3516:
3513:
3510:
3506:
3493:
3489:
3486:generalizing
3485:
3460:
3454:
3451:
3445:
3442:
3427:
3413:
3410:
3404:
3401:
3395:
3392:
3386:
3381:
3377:
3373:
3370:
3355:
3341:
3338:
3332:
3329:
3323:
3318:
3314:
3310:
3307:
3292:
3274:
3271:
3265:
3262:
3259:
3253:
3247:
3244:
3236:
3230:
3227:
3218:
3215:
3212:
3206:
3203:
3197:
3185:
3181:
3177:
3165:
3161:
3157:
3150:
3148:
3143:
3136:
3130:
3124:
3121:
3113:
3107:
3104:
3095:
3092:
3089:
3083:
3080:
3074:
3062:
3058:
3054:
3042:
3038:
3034:
3027:
3024:
3018:
3015:
3012:
3010:
3005:
2981:
2976:
2973:
2968:
2965:
2959:
2956:
2936:
2922:
2919:
2913:
2910:
2904:
2901:
2889:
2883:
2869:
2866:
2860:
2855:
2851:
2847:
2844:
2832:
2826:
2812:
2804:
2800:
2794:
2791:
2787:
2782:
2775:
2770:
2767:
2763:
2759:
2754:
2749:
2745:
2741:
2736:
2731:
2728:
2724:
2717:
2714:
2711:
2705:
2702:
2693:
2687:
2673:
2665:
2662:
2658:
2652:
2649:
2645:
2640:
2633:
2628:
2624:
2620:
2615:
2610:
2607:
2603:
2599:
2594:
2589:
2586:
2582:
2575:
2572:
2569:
2563:
2560:
2551:
2545:
2544:
2543:
2531:
2527:
2521:
2517:
2503:
2495:
2468:
2463:
2459:
2455:
2453:
2446:
2442:
2435:
2432:
2426:
2423:
2413:
2408:
2404:
2400:
2398:
2391:
2387:
2380:
2377:
2371:
2368:
2358:
2353:
2349:
2345:
2343:
2336:
2332:
2325:
2322:
2316:
2313:
2303:
2300:
2294:
2292:
2287:
2281:
2278:
2268:
2265:
2259:
2257:
2252:
2246:
2243:
2233:
2230:
2224:
2222:
2217:
2211:
2208:
2194:is now known:
2189:
2171:
2166:
2162:
2158:
2150:
2146:
2139:
2136:
2130:
2127:
2118:
2116:
2110:
2107:
2099:
2091:
2087:
2080:
2077:
2070:
2066:
2059:
2056:
2050:
2040:
2037:
2027:
2024:
2017:
2015:
2010:
1986:
1979:
1976:
1972:
1969:
1966:
1963:
1960:
1954:
1951:
1947:
1944:
1941:
1935:
1932:
1926:
1923:
1903:
1885:
1877:
1873:
1867:
1863:
1858:
1851:
1846:
1842:
1838:
1833:
1828:
1824:
1820:
1815:
1810:
1806:
1799:
1796:
1791:
1787:
1780:
1777:
1770:
1762:
1758:
1752:
1748:
1743:
1736:
1731:
1727:
1723:
1718:
1713:
1709:
1705:
1700:
1695:
1691:
1684:
1681:
1676:
1672:
1665:
1662:
1655:
1647:
1643:
1637:
1633:
1628:
1621:
1616:
1612:
1608:
1603:
1598:
1594:
1590:
1585:
1580:
1576:
1569:
1566:
1561:
1557:
1550:
1547:
1519:
1512:
1505:
1496:
1495:
1494:
1491:
1487:
1478:
1461:
1456:
1452:
1448:
1446:
1439:
1435:
1428:
1425:
1419:
1416:
1406:
1401:
1397:
1393:
1391:
1384:
1380:
1373:
1370:
1364:
1361:
1351:
1346:
1342:
1338:
1336:
1329:
1325:
1318:
1315:
1309:
1306:
1296:
1293:
1287:
1285:
1280:
1274:
1271:
1261:
1258:
1252:
1250:
1245:
1239:
1236:
1226:
1223:
1217:
1215:
1210:
1204:
1201:
1186:
1179:
1172:
1165:
1155:
1151:
1147:
1136:
1129:
1122:
1108:
1088:
1080:
1071:
1063:
1057:
1050:
1044:
1034:
1019:
1013:
1004:
996:
992:
979:
971:
967:
958:
952:
946:
940:
933:
927:
921:
906:
900:
891:
883:
879:
866:
858:
854:
845:
839:
833:
827:
820:
814:
808:
807:
806:
803:
793:
784:
778:
772:
766:
754:
745:
734:
723:
712:
708:
698:
649:
643:
594:
588:
587:
586:
583:
580:
569:
565:
561:
554:
550:
546:
539:
536:
532:
528:
506:
497:
480:
469:
465:
459:
455:
451:
445:
441:
434:
423:
412:
405:
395:
391:
386:angle of the
384:
373:
366:
360:
359:
358:
347:
343:
332:
315:
286:
255:
254:
253:
251:
243:
234:
232:
220:
213:
200:
193:
183:
180:
176:
158:
155:
150:
118:
104:
90:
76:
63:
55:
53:
49:
45:
40:
34:
30:
26:
25:Weber problem
22:
4244:
4229:
4223:
4216:
4212:
4205:
4197:
4190:
4183:
4173:
4166:
4162:32, 467–486.
4159:
4138:
4133:
4128:32, 467–486.
4125:
4116:
4108:
4103:
4095:
4090:
4082:
4077:
4069:
4063:
4055:
4050:
4042:
4038:
4033:
4025:
4020:
4012:
4007:
3999:
3994:
3985:
3981:
3971:
3952:Paul Krugman
3945:
3936:
3925:
3917:
3910:
3587:
3583:
3574:
3480:
2887:
2830:
2691:
2549:
2529:
2525:
2519:
2515:
2512:
1514:
1507:
1500:
1492:
1488:
1479:
1187:
1174:
1167:
1160:
1153:
1149:
1145:
1131:
1124:
1117:
1106:
1093:
1068:
1061:
1055:
1048:
1042:
1017:
1011:
1002:
994:
990:
977:
969:
965:
956:
950:
944:
938:
931:
925:
904:
898:
889:
881:
877:
864:
856:
852:
843:
837:
831:
825:
818:
812:
804:
791:
782:
776:
770:
764:
759:
732:
721:
717:
710:
706:
647:
592:
584:
578:
567:
563:
559:
552:
548:
544:
540:
534:
530:
526:
502:
478:
467:
463:
453:
449:
439:
432:
421:
410:
403:
389:
382:
371:
364:
352:
345:
341:
330:
313:
248:
218:
211:
198:
191:
184:
179:Alfred Weber
159:
134:
48:Fermat point
41:
29:Alfred Weber
24:
18:
4292:Categories
4149:References
3988:: 355–386.
3428:Determine
3356:Determine
3293:Determine
1009:, and the
896:, and the
681:, and the
626:, and the
484:AD, BD, CD
355:AD, BD, CD
327:, all the
4251:EMS Press
4169:4, 21–34.
4111:4, 21–34.
4028:, London.
4002:4, 21–34.
3876:−
3828:∑
3804:−
3744:∑
3687:‖
3680:−
3667:‖
3661:‖
3648:−
3635:‖
3602:∑
3557:‖
3551:−
3538:‖
3507:∑
3458:∠
3455:−
3449:∠
3440:∠
3408:∠
3405:−
3399:∠
3396:−
3390:∠
3387:−
3382:∘
3368:∠
3336:∠
3333:−
3327:∠
3324:−
3319:∘
3305:∠
3269:∠
3266:
3260:−
3251:∠
3248:
3234:∠
3231:−
3225:∠
3219:
3210:∠
3207:
3198:×
3192:¯
3172:¯
3128:∠
3125:
3111:∠
3108:−
3102:∠
3096:
3087:∠
3084:
3075:×
3069:¯
3049:¯
3028:−
3022:∠
3019:
2963:∠
2960:
2917:∠
2914:−
2908:∠
2899:∠
2864:∠
2861:−
2856:∘
2842:∠
2760:−
2718:−
2709:∠
2706:
2621:−
2576:−
2567:∠
2564:
2464:∘
2443:α
2439:∠
2430:∠
2421:∠
2409:∘
2388:α
2384:∠
2375:∠
2366:∠
2354:∘
2333:α
2329:∠
2320:∠
2311:∠
2298:∠
2285:∠
2276:∠
2263:∠
2250:∠
2241:∠
2228:∠
2215:∠
2206:∠
2167:∘
2159:−
2147:α
2143:∠
2134:∠
2125:∠
2088:α
2084:∠
2081:
2067:α
2063:∠
2060:
2051:×
2045:¯
2032:¯
1973:
1948:
1930:∠
1927:
1839:−
1800:−
1788:α
1784:∠
1781:
1724:−
1685:−
1673:α
1669:∠
1666:
1609:−
1570:−
1558:α
1554:∠
1551:
1457:∘
1436:α
1432:∠
1423:∠
1414:∠
1402:∘
1381:α
1377:∠
1368:∠
1359:∠
1347:∘
1326:α
1322:∠
1313:∠
1304:∠
1291:∠
1278:∠
1269:∠
1256:∠
1243:∠
1234:∠
1221:∠
1208:∠
1199:∠
304:; on arc
3490:for the
2111:′
1980:′
1955:′
725:, where
571:, where
471:, where
368:, whose
21:geometry
4253:, 2001
1183:A, B, C
1101:A, B, C
573:E, F, G
282:A, B, C
262:A, B, C
166:A, B, C
154:Tellier
141:A, B, C
2994:where
1999:where
984:, the
871:, the
666:, the
651:, the
611:, the
596:, the
310:, any
256:Point
23:, the
3963:Notes
3911:The
2507:D, E
1052:and
789:and
517:A, B
509:A, B
473:F, G
440:AD'B
377:ABDE
205:and
3590:+ 1
3378:180
3315:180
3263:cos
3245:sin
3216:cos
3204:sin
3122:sin
3093:sin
3081:sin
3016:sin
2957:tan
2852:180
2703:cos
2561:cos
2460:180
2405:180
2350:180
2163:180
2078:sin
2057:sin
1970:cos
1945:sin
1924:tan
1778:cos
1663:cos
1548:cos
1531:, w
1527:, w
1513:, ∠
1506:, ∠
1453:180
1398:180
1343:180
1173:, ∠
1166:, ∠
1152:, ∠
1148:, ∠
1130:, ∠
1123:, ∠
1107:ABC
733:ABC
722:ACF
711:BCG
709:, △
707:ABE
648:BCG
593:ABE
579:ABC
568:BCG
566:, △
564:ACF
562:, △
560:ABE
553:BCG
551:, △
549:ACF
547:, △
545:ABE
535:BDC
533:, ∠
531:ADC
529:, ∠
527:ADB
519:or
511:or
479:ABC
468:BCG
466:, △
464:ACF
454:BCD
452:, △
450:ACD
433:ABC
422:ABC
411:ABE
404:ADB
390:ABE
383:ABE
372:ADB
365:ABD
346:AjB
344:, ∠
342:AiB
331:AjB
324:AjB
314:AiB
307:AiB
301:AjB
297:AiB
19:In
4294::
4249:,
4243:,
3986:43
3980:.
3954:a
3461:5.
3430:∠2
3358:∠5
3295:∠1
2949:):
2939:∠3
2524:,
2192:∠3
1916:):
1906:∠3
1056:RA
1043:RA
986:RI
957:RA
926:RA
873:RH
844:RA
813:RA
744:.
738:AC
683:CG
668:BG
653:BC
628:BE
613:AE
598:AB
427:D'
299:,
291:AB
217:,
197:,
3890:|
3884:j
3880:y
3871:i
3867:x
3862:|
3855:i
3851:w
3843:n
3838:1
3835:=
3832:i
3818:|
3812:j
3808:y
3799:i
3795:x
3790:|
3782:i
3778:x
3772:i
3768:w
3759:n
3754:1
3751:=
3748:i
3737:=
3732:1
3729:+
3726:j
3722:y
3709:i
3707:w
3691:2
3683:y
3675:i
3671:x
3656:j
3652:y
3643:i
3639:x
3629:i
3625:w
3617:n
3612:1
3609:=
3606:i
3588:j
3584:y
3578:0
3575:y
3560:,
3554:y
3546:i
3542:x
3532:i
3528:w
3522:n
3517:1
3514:=
3511:i
3496:y
3452:a
3446:=
3443:2
3432::
3414:;
3411:1
3402:c
3393:b
3374:=
3371:5
3360::
3342:;
3339:3
3330:e
3311:=
3308:1
3297::
3275:;
3272:f
3254:c
3240:)
3237:b
3228:e
3222:(
3213:d
3186:2
3182:A
3178:R
3166:1
3162:A
3158:R
3151:=
3144:y
3137:;
3131:c
3117:)
3114:b
3105:e
3099:(
3090:d
3063:2
3059:A
3055:R
3043:1
3039:A
3035:R
3025:f
3013:=
3006:x
2982:;
2977:y
2974:x
2969:=
2966:3
2947:E
2943:D
2923:;
2920:c
2911:e
2905:=
2902:d
2891::
2888:d
2886:∠
2870:;
2867:p
2848:=
2845:c
2834::
2831:c
2829:∠
2813:;
2805:R
2801:w
2795:1
2792:A
2788:w
2783:2
2776:2
2771:2
2768:A
2764:w
2755:2
2750:R
2746:w
2742:+
2737:2
2732:1
2729:A
2725:w
2715:=
2712:p
2695::
2692:p
2690:∠
2674:;
2666:2
2663:A
2659:w
2653:1
2650:A
2646:w
2641:2
2634:2
2629:R
2625:w
2616:2
2611:2
2608:A
2604:w
2600:+
2595:2
2590:1
2587:A
2583:w
2573:=
2570:e
2553::
2550:e
2548:∠
2539:R
2537:w
2532:2
2530:A
2526:w
2522:1
2520:A
2516:w
2509:.
2469:.
2456:=
2447:C
2436:+
2433:5
2427:+
2424:4
2414:;
2401:=
2392:B
2381:+
2378:3
2372:+
2369:2
2359:;
2346:=
2337:A
2326:+
2323:6
2317:+
2314:1
2304:;
2301:B
2295:=
2288:6
2282:+
2279:5
2269:;
2266:A
2260:=
2253:4
2247:+
2244:3
2234:;
2231:C
2225:=
2218:2
2212:+
2209:1
2172:.
2156:)
2151:C
2140:+
2137:B
2131:+
2128:A
2122:(
2119:=
2108:k
2100:,
2092:A
2071:B
2041:A
2038:C
2028:B
2025:C
2018:=
2011:k
1987:;
1977:k
1967:k
1964:+
1961:1
1952:k
1942:k
1936:=
1933:3
1914:E
1910:D
1886:;
1878:B
1874:w
1868:A
1864:w
1859:2
1852:2
1847:C
1843:w
1834:2
1829:B
1825:w
1821:+
1816:2
1811:A
1807:w
1797:=
1792:C
1771:;
1763:C
1759:w
1753:A
1749:w
1744:2
1737:2
1732:B
1728:w
1719:2
1714:C
1710:w
1706:+
1701:2
1696:A
1692:w
1682:=
1677:B
1656:;
1648:C
1644:w
1638:B
1634:w
1629:2
1622:2
1617:A
1613:w
1604:2
1599:C
1595:w
1591:+
1586:2
1581:B
1577:w
1567:=
1562:A
1533:C
1529:B
1525:A
1523:w
1517:C
1515:α
1510:B
1508:α
1503:A
1501:α
1499:∠
1483:P
1462:.
1449:=
1440:C
1429:+
1426:5
1420:+
1417:4
1407:;
1394:=
1385:B
1374:+
1371:3
1365:+
1362:2
1352:;
1339:=
1330:A
1319:+
1316:6
1310:+
1307:1
1297:;
1294:B
1288:=
1281:6
1275:+
1272:5
1262:;
1259:A
1253:=
1246:4
1240:+
1237:3
1227:;
1224:C
1218:=
1211:2
1205:+
1202:1
1177:C
1175:α
1170:B
1168:α
1163:A
1161:α
1159:∠
1154:C
1150:B
1146:A
1144:∠
1134:C
1132:α
1127:B
1125:α
1120:A
1118:α
1116:∠
1105:△
1097:P
1062:I
1059:1
1054:△
1049:H
1046:2
1041:△
1037:D
1032:;
1030:R
1025:R
1023:w
1018:I
1015:1
1012:A
1006:1
1003:A
997:1
995:A
991:w
981:2
978:A
972:2
970:A
966:w
960:1
951:R
948:2
945:A
942:1
939:A
937:△
932:I
929:1
924:△
919:;
917:R
912:R
910:w
905:H
902:2
899:A
893:2
890:A
884:2
882:A
878:w
868:1
865:A
859:1
857:A
853:w
847:2
838:R
835:2
832:A
829:1
826:A
824:△
819:H
816:2
811:△
800:R
795:2
792:A
786:1
783:A
777:R
774:2
771:A
768:1
765:A
763:△
742:D
731:△
727:F
720:△
705:△
701:D
696:;
694:C
689:B
687:w
679:B
674:C
672:w
664:A
659:A
657:w
646:△
641:;
639:A
634:A
632:w
624:B
619:B
617:w
609:C
604:C
602:w
591:△
577:△
558:△
543:△
525:∠
521:C
513:C
477:△
462:△
457:;
448:△
438:∠
431:△
420:△
416:E
409:△
402:∠
398:D
388:△
381:∠
370:∠
363:△
340:∠
336:j
329:∠
319:i
312:∠
284:;
278:D
274:C
270:B
266:A
258:D
227:R
222:2
219:A
215:1
212:A
207:R
202:2
199:A
195:1
192:A
187:D
170:D
162:D
145:D
137:D
37:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
