36:
4409:
3418:
3413:
421:
position of the plumbline is traced on the surface, and the procedure is repeated with the pin inserted at any different point (or a number of points) off the centroid of the object. The unique intersection point of these lines will be the centroid (figure c). Provided that the body is of uniform density, all lines made this way will include the centroid, and all lines will cross at exactly the same place.
93:
4872:
447:
438:
429:
3237:
4047:
3101:
4630:
2831:
5058:
1438:
239:
The center of gravity, as the name indicates, is a notion that arose in mechanics, most likely in connection with building activities. It is uncertain when the idea first appeared, as the concept likely occurred to many people individually with minor differences. Nonetheless, the center of gravity of
5073:
Any of the three medians through the centroid divides the triangle's area in half. This is not true for other lines through the centroid; the greatest departure from the equal-area division occurs when a line through the centroid is parallel to a side of the triangle, creating a smaller triangle and
468:
This method can be extended (in theory) to concave shapes where the centroid may lie outside the shape, and virtually to solids (again, of uniform density), where the centroid may lie within the body. The (virtual) positions of the plumb lines need to be recorded by means other than by drawing them
420:
and a pin to find the collocated center of mass of a thin body of uniform density having the same shape. The body is held by the pin, inserted at a point, off the presumed centroid in such a way that it can freely rotate around the pin; the plumb line is then dropped from the pin (figure b). The
4404:{\displaystyle {\begin{aligned}C&={\frac {1}{a}}:{\frac {1}{b}}:{\frac {1}{c}}=bc:ca:ab=\csc L:\csc M:\csc N\\&=\cos L+\cos M\cdot \cos N:\cos M+\cos N\cdot \cos L:\cos N+\cos L\cdot \cos M\\&=\sec L+\sec M\cdot \sec N:\sec M+\sec N\cdot \sec L:\sec N+\sec L\cdot \sec M.\end{aligned}}}
477:
For convex two-dimensional shapes, the centroid can be found by balancing the shape on a smaller shape, such as the top of a narrow cylinder. The centroid occurs somewhere within the range of contact between the two shapes (and exactly at the point where the shape would balance on a pin). In
4413:
The centroid is also the physical center of mass if the triangle is made from a uniform sheet of material; or if all the mass is concentrated at the three vertices, and evenly divided among them. On the other hand, if the mass is distributed along the triangle's perimeter, with uniform
2401:
1013:
3902:
478:
principle, progressively narrower cylinders can be used to find the centroid to arbitrary precision. In practice air currents make this infeasible. However, by marking the overlap range from multiple balances, one can achieve a considerable level of accuracy.
4867:{\displaystyle {\begin{aligned}{\overline {CH}}&=4{\overline {CN}},\\{\overline {CO}}&=2{\overline {CN}},\\{\overline {IC}}&<{\overline {HC}},\\{\overline {IH}}&<{\overline {HC}},\\{\overline {IC}}&<{\overline {IO}}.\end{aligned}}}
6404:
6221:
4930:
1282:
683:
3096:{\displaystyle {\begin{aligned}{\bar {x}}&={\frac {1}{A}}\int _{a}^{b}x{\bigl (}f(x)-g(x){\bigr )}\,dx,\\{\bar {y}}&={\frac {1}{A}}\int _{a}^{b}{\tfrac {1}{2}}{\bigl (}f(x)+g(x){\bigr )}{\bigl (}f(x)-g(x){\bigr )}\,dx,\end{aligned}}}
2262:
6562:
834:
4635:
3272:
Divide the shape into two other rectangles, as shown in fig 3. Find the centroids of these two rectangles by drawing the diagonals. Draw a line joining the centroids. The centroid of the L-shape must lie on this line
569:
3720:
5399:
3243:
Divide the shape into two rectangles, as shown in fig 2. Find the centroids of these two rectangles by drawing the diagonals. Draw a line joining the centroids. The centroid of the shape must lie on this line
2066:
4052:
3963:
5525:
2836:
1718:
7138:
of the way from the center to the pole). The centroid of a hollow hemisphere (i.e. half of a hollow sphere) divides the line segment connecting the sphere's center to the hemisphere's pole in half.
7045:
6230:
6047:
3202:
4574:
182:
coincides with the centroid. Informally, it can be understood as the point at which a cutout of the shape (with uniformly distributed mass) could be perfectly balanced on the tip of a pin.
5794:
603:
1854:
5931:
2671:
773:
6964:
2567:
2483:
5991:
1756:
1530:
5875:
3669:
3614:
3560:
1634:
598:
7136:
6812:
6783:
5105:
4460:
3497:
6659:
6040:
6610:
2824:
2758:
293:
7549:
Gerald A. Edgar, Daniel H. Ullman & Douglas B. West (2018) Problems and
Solutions, The American Mathematical Monthly, 125:1, 81-89, DOI: 10.1080/00029890.2018.1397465
3220:) can be used to find the centroid of an object of irregular shape with smooth (or piecewise smooth) boundary. The mathematical principle involved is a special case of
5652:
248:(287–212 BCE) with being the first to find the centroid of plane figures, although he never defines it. A treatment of centroids of solids by Archimedes has been lost.
6437:
2230:
2174:
1915:
1880:
1498:
1068:
5608:
5160:
4039:
4006:
2095:
1468:
1122:
1095:
827:
800:
6874:
6711:
5701:
5053:{\displaystyle ({\text{Area of }}\triangle ABG)=({\text{Area of }}\triangle ACG)=({\text{Area of }}\triangle BCG)={\tfrac {1}{3}}({\text{Area of }}\triangle ABC).}
4923:
3468:
1433:{\displaystyle x={\frac {5\times 10^{2}+13.33\times {\frac {1}{2}}10^{2}-3\times \pi 2.5^{2}}{10^{2}+{\frac {1}{2}}10^{2}-\pi 2.5^{2}}}\approx 8.5{\text{ units}}.}
7107:
7073:
6685:
5556:
3348:
3297:
3268:
6734:
5183:
4625:
4516:
3371:
3322:
2613:
2529:
2446:
2253:
2138:
1942:
1553:
1205:
1038:
6909:
6426:
5825:
5672:
5576:
5264:
5243:
5223:
5203:
5128:
4894:
4599:
4490:
3709:
3689:
3391:
3123:
2793:
2714:
2694:
2587:
2503:
2423:
2194:
2115:
1962:
1800:
1776:
1605:
1573:
1279:(c). Then the centroid of the figure is the weighted average of the three points. The horizontal position of the centroid, from the left edge of the figure is
1225:
1182:
1162:
1142:
711:
504:
150:
7083:
The centroid of a solid hemisphere (i.e. half of a solid ball) divides the line segment connecting the sphere's center to the hemisphere's pole in the ratio
7049:
The geometric centroid coincides with the center of mass if the mass is uniformly distributed over the whole simplex, or concentrated at the vertices as
5272:
1972:
1230:
For example, the figure below (a) is easily divided into a square and a triangle, both with positive area; and a circular hole, with negative area (b).
251:
It is unlikely that
Archimedes learned the theorem that the medians of a triangle meet in a point—the center of gravity of the triangle—directly from
230:" when the purely geometrical aspects of that point are to be emphasized. The term is peculiar to the English language; French, for instance, uses "
5408:
7679:
509:
6566:
In these formulae, the vertices are assumed to be numbered in order of their occurrence along the polygon's perimeter; furthermore, the vertex
1641:
2396:{\displaystyle C_{\mathrm {x} }={\frac {\int xS_{\mathrm {y} }(x)\ dx}{A}},\quad C_{\mathrm {y} }={\frac {\int yS_{\mathrm {x} }(y)\ dy}{A}},}
6975:
3966:
269:. It may be added, in passing, that the proposition did not become common in the textbooks on plane geometry until the nineteenth century.
1008:{\displaystyle C_{x}={\frac {\sum _{i}{C_{i}}_{x}A_{i}}{\sum _{i}A_{i}}},\quad C_{y}={\frac {\sum _{i}{C_{i}}_{y}A_{i}}{\sum _{i}A_{i}}}.}
57:
4523:
5403:
The sum of the squares of the triangle's sides equals three times the sum of the squared distances of the centroid from the vertices:
7429:
3909:
1263:
1251:
1239:
79:
5707:
5529:
A triangle's centroid is the point that maximizes the product of the directed distances of a point from the triangle's sidelines.
281:
object always lies in the object. A non-convex object might have a centroid that is outside the figure itself. The centroid of a
3897:{\displaystyle C={\tfrac {1}{3}}(L+M+N)={\bigl (}{\tfrac {1}{3}}(x_{L}+x_{M}+x_{N}),{\tfrac {1}{3}}(y_{L}+y_{M}+y_{N}){\bigr )}.}
3128:
6785:
the distance from the base to the apex. For a cone or pyramid that is just a shell (hollow) with no base, the centroid is
226:
The term "centroid" is of recent coinage (1814). It is used as a substitute for the older terms "center of gravity" and "
7181:
50:
44:
7776:
7631:
7228:
7786:
1805:
7572:
Leung, Kam-tim; and Suen, Suk-nam; "Vectors, matrices and geometry", Hong Kong
University Press, 1994, pp. 53–54
6399:{\displaystyle C_{\mathrm {y} }={\frac {1}{6A}}\sum _{i=0}^{n-1}(y_{i}+y_{i+1})(x_{i}\ y_{i+1}-x_{i+1}\ y_{i}),}
6216:{\displaystyle C_{\mathrm {x} }={\frac {1}{6A}}\sum _{i=0}^{n-1}(x_{i}+x_{i+1})(x_{i}\ y_{i+1}-x_{i+1}\ y_{i}),}
5879:
61:
7771:
7651:
2625:
716:
7732:
6918:
6831:
1040:
overlaps between the parts, or parts that extend outside the figure can all be handled using negative areas
6736:
computed as above, will be negative; however, the centroid coordinates will be correct even in this case.)
371:
2534:
2450:
5935:
3973:
3500:
1918:
1576:
282:
1729:
1503:
7688:
6838:. A line segment joining a vertex of a tetrahedron with the centroid of the opposite face is called a
5830:
3619:
3564:
3510:
1610:
685:
This point minimizes the sum of squared
Euclidean distances between itself and each point in the set.
574:
214:, the centroid of a radial projection of a region of the Earth's surface to sea level is the region's
132:
position of all the points in the surface of the figure. The same definition extends to any object in
7112:
6788:
6759:
5081:
4436:
3473:
678:{\displaystyle \mathbf {C} ={\frac {\mathbf {x} _{1}+\mathbf {x} _{2}+\cdots +\mathbf {x} _{k}}{k}}.}
257:
6615:
5996:
7751:
7529:"Trilinear distance inequalities for the symmedian point, the centroid, and other triangle centers"
6569:
2797:
2719:
2674:
317:
215:
7309:
7262:
6749:
5063:
1779:
313:
262:
3221:
5613:
7707:
7667:
7635:
7594:
7590:
7301:
7171:
7163:
6967:
6846:. Hence there are four medians and three bimedians. These seven line segments all meet at the
4602:
3712:
3435:
3431:
2199:
2143:
1276:
383:
374:
is undefined (or lies outside the enclosing space), because a translation has no fixed point.
333:
329:
191:
7528:
4462:
times the length of any side times the perpendicular distance from the side to the centroid.
7293:
7254:
7148:
6429:
1885:
1859:
1473:
1097:
should be taken with positive and negative signs in such a way that the sum of the signs of
1043:
7724:
5581:
5133:
4012:
3979:
3976:
the centroid can be expressed in any of these equivalent ways in terms of the side lengths
2073:
1446:
1100:
1073:
805:
778:
7433:
6853:
6835:
6753:
6745:
6690:
5677:
4899:
4430:), which does not (in general) coincide with the geometric centroid of the full triangle.
4427:
3504:
3444:
3403:
386:
of the triangle (each median connecting a vertex with the midpoint of the opposite side).
309:
204:
156:
129:
7755:
7158:
7086:
7052:
6664:
5535:
3438:
with the midpoint of the opposite side). The centroid divides each of the medians in the
3327:
3276:
3247:
300:. In particular, the geometric centroid of an object lies in the intersection of all its
6716:
5165:
4607:
4498:
3353:
3304:
2595:
2511:
2428:
2235:
2120:
1924:
1535:
1187:
1020:
7781:
7710:
7663:
7587:
College
Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle
6894:
6411:
5810:
5657:
5561:
5249:
5228:
5208:
5188:
5113:
4879:
4584:
4475:
4419:
4415:
3694:
3674:
3376:
3108:
2763:
2699:
2679:
2572:
2488:
2408:
2179:
2100:
1947:
1785:
1761:
1590:
1558:
1210:
1167:
1147:
1127:
696:
489:
402:
349:
297:
241:
227:
178:
135:
7765:
7313:
7153:
6880:
and circumcenter (center of the circumscribed sphere). These three points define the
413:
364:
356:
345:
286:
196:
7758:, an interactive dynamic geometry sketch using the gravity simulator of Cinderella.
7742:
7736:
7233:
4493:
3499:
of the distance from each side to the opposite vertex (see figures at right). Its
3417:
3412:
167:
125:
121:
6557:{\displaystyle A={\frac {1}{2}}\sum _{i=0}^{n-1}(x_{i}\ y_{i+1}-x_{i+1}\ y_{i}).}
5205:
from the three vertices exceeds the sum of the squared distances of the centroid
7746:
6877:
6827:
4470:
101:
92:
446:
437:
428:
6885:
4466:
3217:
301:
278:
245:
172:
7606:
7305:
6842:, and a line segment joining the midpoints of two opposite edges is called a
3507:
of the coordinates of the three vertices. That is, if the three vertices are
7715:
5075:
5067:
3236:
3213:
417:
341:
321:
211:
200:
153:
1443:
The same formula holds for any three-dimensional objects, except that each
416:, such as in figure (a) below, may be determined experimentally by using a
6756:
to the centroid of the base. For a solid cone or pyramid, the centroid is
564:{\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{k}}
17:
7258:
6850:
of the tetrahedron. The medians are divided by the centroid in the ratio
4579:
4423:
3427:
3407:
2590:
2506:
1723:
360:
305:
163:
7266:
7491:
7297:
6912:
5804:
713:
can be computed by dividing it into a finite number of simpler figures
337:
325:
236:" on most occasions, and other languages use terms of similar meaning.
186:
105:
7281:
7176:
252:
3232:
This is a method of determining the centroid of an L-shaped object.
7646:
Larson, Roland E.; Hostetler, Robert P.; Edwards, Bruce H. (1998),
2257:
For a plane figure, in particular, the barycentric coordinates are
7619:
5394:{\displaystyle PA^{2}+PB^{2}+PC^{2}=GA^{2}+GB^{2}+GC^{2}+3PG^{2}.}
3439:
2061:{\displaystyle C_{k}={\frac {\int zS_{k}(z)\ dz}{\int g(x)\ dx}},}
91:
7671:
7639:
7598:
1440:
The vertical position of the centroid is found in the same way.
1262:
1250:
1238:
7282:"Archimedes' lost treatise on the centers of gravity of solids"
3958:{\displaystyle {\tfrac {1}{3}}:{\tfrac {1}{3}}:{\tfrac {1}{3}}}
5225:
from the vertices by three times the squared distance between
29:
7752:
Experimentally finding the medians and centroid of a triangle
6915:
in the following way. If the set of vertices of a simplex is
5520:{\displaystyle AB^{2}+BC^{2}+CA^{2}=3(GA^{2}+GB^{2}+GC^{2}).}
261:. The first explicit statement of this proposition is due to
445:
436:
427:
382:
The centroid of a triangle is the intersection of the three
7245:
Court, Nathan
Altshiller (1960). "Notes on the centroid".
6876:
The centroid of a tetrahedron is the midpoint between its
7229:
6713:(If the points are numbered in clockwise order, the area
1713:{\displaystyle C={\frac {\int xg(x)\ dx}{\int g(x)\ dx}}}
7040:{\displaystyle C={\frac {1}{n+1}}\sum _{i=0}^{n}v_{i}.}
4518:
exactly twice as close to the latter as to the former:
3197:{\textstyle \int _{a}^{b}{\bigl (}f(x)-g(x){\bigr )}dx}
1500:
rather than its area. It also holds for any subset of
7117:
6793:
6764:
5130:
be any point in the plane of a triangle with vertices
5086:
5013:
4441:
3944:
3929:
3914:
3831:
3774:
3731:
3478:
3350:
it must be at the intersection of these two lines, at
3131:
2986:
7729:
by Clark
Kimberling. The centroid is indexed as X(2).
7115:
7089:
7055:
6978:
6921:
6897:
6856:
6791:
6762:
6719:
6693:
6667:
6618:
6572:
6440:
6414:
6233:
6050:
5999:
5938:
5882:
5833:
5813:
5710:
5680:
5660:
5616:
5584:
5564:
5538:
5411:
5275:
5252:
5231:
5211:
5191:
5168:
5136:
5116:
5084:
4933:
4902:
4882:
4633:
4610:
4587:
4526:
4501:
4478:
4439:
4050:
4015:
3982:
3912:
3723:
3697:
3677:
3622:
3567:
3513:
3476:
3447:
3379:
3356:
3330:
3307:
3279:
3250:
3111:
2834:
2800:
2766:
2722:
2702:
2682:
2628:
2598:
2575:
2537:
2514:
2491:
2453:
2431:
2411:
2265:
2238:
2202:
2182:
2146:
2123:
2103:
2076:
1975:
1950:
1927:
1888:
1862:
1808:
1788:
1764:
1732:
1644:
1613:
1593:
1561:
1538:
1506:
1476:
1449:
1285:
1213:
1190:
1170:
1150:
1130:
1103:
1076:
1046:
1023:
837:
808:
781:
719:
699:
606:
577:
512:
492:
138:
1917:
otherwise. Note that the denominator is simply the
370:
For the same reason, the centroid of an object with
7747:
Centroid construction with compass and straightedge
7658:Protter, Murray H.; Morrey, Charles B. Jr. (1970),
4569:{\displaystyle {\overline {CH}}=2{\overline {CO}}.}
389:For other properties of a triangle's centroid, see
352:, etc.) can be determined by this principle alone.
7130:
7101:
7067:
7039:
6958:
6903:
6868:
6806:
6777:
6728:
6705:
6679:
6653:
6604:
6556:
6428:is the polygon's signed area, as described by the
6420:
6398:
6215:
6034:
5985:
5925:
5869:
5819:
5788:
5695:
5666:
5646:
5602:
5570:
5550:
5519:
5393:
5258:
5237:
5217:
5197:
5177:
5154:
5122:
5099:
5052:
4917:
4888:
4866:
4619:
4593:
4568:
4510:
4484:
4454:
4403:
4033:
4000:
3957:
3896:
3703:
3683:
3663:
3608:
3554:
3491:
3462:
3385:
3365:
3342:
3316:
3291:
3262:
3196:
3117:
3095:
2818:
2787:
2752:
2708:
2688:
2665:
2607:
2581:
2561:
2523:
2497:
2477:
2440:
2417:
2395:
2247:
2224:
2188:
2168:
2132:
2109:
2089:
2060:
1964:has zero measure, or if either integral diverges.
1956:
1936:
1909:
1874:
1848:
1794:
1770:
1750:
1712:
1628:
1599:
1567:
1547:
1524:
1492:
1462:
1432:
1219:
1199:
1176:
1156:
1136:
1116:
1089:
1062:
1032:
1007:
821:
794:
767:
705:
677:
592:
563:
498:
289:, for example, lies in the object's central void.
144:
6752:is located on the line segment that connects the
1821:
265:(perhaps the first century CE) and occurs in his
7607:"Calculating the area and centroid of a polygon"
7514:
7465:
7403:
7329:
3393:might lie inside or outside the L-shaped object.
2232:Again, the denominator is simply the measure of
7681:Locating the centre of mass by mechanical means
7510:
7508:
5803:The centroid of a non-self-intersecting closed
7325:
7323:
6814:the distance from the base plane to the apex.
1275:The centroid of each part can be found in any
240:figures was studied extensively in Antiquity;
7427:Clark Kimberling's Encyclopedia of Triangles
7368:
7356:
7344:
7215:
7203:
5789:{\displaystyle PA+PB+PC\leq 2(PD+PE+PF)+3PG.}
3886:
3768:
3183:
3149:
3074:
3040:
3033:
2999:
2922:
2888:
8:
7340:
7338:
7199:
7197:
6884:of the tetrahedron that is analogous to the
3301:As the centroid of the shape must lie along
2196:with the hyperplane defined by the equation
231:
1849:{\displaystyle \mathbb {R} ^{n}\!:\ g(x)=1}
1256:(b) Object described using simpler elements
7492:"Medians and Area Bisectors of a Triangle"
5926:{\displaystyle (x_{1},y_{1}),\;\ldots ,\;}
5922:
5915:
5866:
2647:
1944:This formula cannot be applied if the set
27:Mean position of all the points in a shape
7557:
7555:
7116:
7114:
7088:
7054:
7028:
7018:
7007:
6985:
6977:
6946:
6927:
6922:
6920:
6896:
6855:
6792:
6790:
6763:
6761:
6718:
6692:
6666:
6639:
6626:
6617:
6593:
6580:
6571:
6542:
6523:
6504:
6491:
6472:
6461:
6447:
6439:
6413:
6384:
6365:
6346:
6333:
6311:
6298:
6279:
6268:
6249:
6239:
6238:
6232:
6201:
6182:
6163:
6150:
6128:
6115:
6096:
6085:
6066:
6056:
6055:
6049:
6020:
6007:
5998:
5965:
5946:
5937:
5903:
5890:
5881:
5854:
5841:
5832:
5812:
5709:
5679:
5659:
5615:
5583:
5563:
5537:
5505:
5489:
5473:
5451:
5435:
5419:
5410:
5382:
5363:
5347:
5331:
5315:
5299:
5283:
5274:
5251:
5230:
5210:
5190:
5185:Then the sum of the squared distances of
5167:
5135:
5115:
5085:
5083:
5027:
5012:
4989:
4963:
4937:
4932:
4901:
4881:
4842:
4820:
4798:
4776:
4754:
4732:
4710:
4685:
4663:
4638:
4634:
4632:
4609:
4586:
4548:
4527:
4525:
4500:
4477:
4440:
4438:
4091:
4078:
4065:
4051:
4049:
4014:
3981:
3943:
3928:
3913:
3911:
3885:
3884:
3875:
3862:
3849:
3830:
3818:
3805:
3792:
3773:
3767:
3766:
3730:
3722:
3696:
3676:
3649:
3636:
3621:
3594:
3581:
3566:
3540:
3527:
3512:
3477:
3475:
3446:
3378:
3355:
3329:
3306:
3278:
3249:
3182:
3181:
3148:
3147:
3141:
3136:
3130:
3110:
3079:
3073:
3072:
3039:
3038:
3032:
3031:
2998:
2997:
2985:
2979:
2974:
2960:
2942:
2941:
2927:
2921:
2920:
2887:
2886:
2877:
2872:
2858:
2840:
2839:
2835:
2833:
2799:
2765:
2721:
2701:
2681:
2673:of a region bounded by the graphs of the
2666:{\displaystyle ({\bar {x}},\;{\bar {y}})}
2649:
2648:
2633:
2632:
2627:
2597:
2574:
2543:
2542:
2536:
2513:
2490:
2459:
2458:
2452:
2430:
2410:
2359:
2358:
2345:
2335:
2334:
2295:
2294:
2281:
2271:
2270:
2264:
2237:
2207:
2201:
2181:
2151:
2145:
2122:
2102:
2081:
2075:
2002:
1989:
1980:
1974:
1949:
1926:
1887:
1861:
1815:
1811:
1810:
1807:
1787:
1763:
1739:
1735:
1734:
1731:
1651:
1643:
1620:
1616:
1615:
1612:
1592:
1560:
1537:
1513:
1509:
1508:
1505:
1481:
1475:
1454:
1448:
1422:
1407:
1391:
1377:
1368:
1356:
1334:
1320:
1305:
1292:
1284:
1212:
1189:
1169:
1149:
1129:
1124:for all parts that enclose a given point
1108:
1102:
1081:
1075:
1051:
1045:
1022:
993:
983:
971:
961:
954:
949:
942:
935:
926:
909:
899:
887:
877:
870:
865:
858:
851:
842:
836:
813:
807:
786:
780:
768:{\displaystyle X_{1},X_{2},\dots ,X_{n},}
756:
737:
724:
718:
698:
660:
655:
639:
634:
624:
619:
615:
607:
605:
584:
580:
579:
576:
555:
550:
534:
529:
519:
514:
511:
491:
137:
80:Learn how and when to remove this message
43:This article includes a list of general
7660:College Calculus with Analytic Geometry
7453:
7193:
5078:; in this case the trapezoid's area is
1268:(c) Centroids of elements of the object
7561:
7380:
6959:{\displaystyle {v_{0},\ldots ,v_{n}},}
4418:, then the center of mass lies at the
2176:is the measure of the intersection of
6687:on the last case must loop around to
2569:is the length of the intersection of
2485:is the length of the intersection of
7:
3430:is the point of intersection of its
3125:is the area of the region (given by
1967:Another formula for the centroid is
1636:can also be computed by the formula
292:If the centroid is defined, it is a
255:, as this proposition is not in the
7733:Characteristic Property of Centroid
7477:
7415:
7392:
2562:{\displaystyle S_{\mathrm {x} }(y)}
2478:{\displaystyle S_{\mathrm {y} }(x)}
403:Center of mass § Determination
7430:"Encyclopedia of Triangle Centers"
6240:
6057:
5986:{\displaystyle (x_{n-1},y_{n-1}),}
5032:
4994:
4968:
4942:
4465:A triangle's centroid lies on its
2544:
2460:
2360:
2336:
2296:
2272:
1277:list of centroids of simple shapes
412:The centroid of a uniformly dense
390:
49:it lacks sufficient corresponding
25:
7585:Altshiller-Court, Nathan (1925),
6966:then considering the vertices as
1751:{\displaystyle \mathbb {R} ^{n},}
1525:{\displaystyle \mathbb {R} ^{d},}
829:of each part, and then computing
355:In particular, the centroid of a
308:. The centroid of many figures (
7726:Encyclopedia of Triangle Centers
6891:These results generalize to any
5870:{\displaystyle (x_{0},y_{0}),\;}
5066:of a triangle's centroid is its
4896:is the centroid of the triangle
3664:{\displaystyle N=(x_{N},y_{N}),}
3609:{\displaystyle M=(x_{M},y_{M}),}
3555:{\displaystyle L=(x_{L},y_{L}),}
3416:
3411:
3235:
1629:{\displaystyle \mathbb {R} ^{n}}
1261:
1249:
1237:
656:
635:
620:
608:
593:{\displaystyle \mathbb {R} ^{n}}
551:
530:
515:
486:The centroid of a finite set of
423:
359:is the meeting point of its two
34:
7741:Interactive animations showing
7131:{\displaystyle {\tfrac {3}{8}}}
6807:{\displaystyle {\tfrac {1}{3}}}
6778:{\displaystyle {\tfrac {1}{4}}}
5107:that of the original triangle.
5100:{\displaystyle {\tfrac {5}{9}}}
4455:{\displaystyle {\tfrac {3}{2}}}
3691:here but most commonly denoted
3492:{\displaystyle {\tfrac {1}{3}}}
2329:
1726:are taken over the whole space
1555:with the areas replaced by the
921:
693:The centroid of a plane figure
7286:The Mathematical Intelligencer
6654:{\displaystyle (x_{0},y_{0}),}
6645:
6619:
6599:
6573:
6548:
6484:
6390:
6326:
6323:
6291:
6207:
6143:
6140:
6108:
6035:{\displaystyle (C_{x},C_{y}),}
6026:
6000:
5977:
5939:
5909:
5883:
5860:
5834:
5768:
5741:
5511:
5463:
5044:
5024:
5006:
4986:
4980:
4960:
4954:
4934:
3881:
3842:
3824:
3785:
3760:
3742:
3655:
3629:
3600:
3574:
3546:
3520:
3470:which is to say it is located
3178:
3172:
3163:
3157:
3069:
3063:
3054:
3048:
3028:
3022:
3013:
3007:
2947:
2917:
2911:
2902:
2896:
2845:
2779:
2767:
2747:
2741:
2732:
2726:
2660:
2654:
2638:
2629:
2556:
2550:
2472:
2466:
2372:
2366:
2308:
2302:
2163:
2157:
2040:
2034:
2014:
2008:
1898:
1892:
1837:
1831:
1695:
1689:
1669:
1663:
1:
7648:Calculus of a Single Variable
6834:having four triangles as its
6612:is assumed to be the same as
6605:{\displaystyle (x_{n},y_{n})}
5610:be the midpoints of segments
3906:The centroid is therefore at
2819:{\displaystyle a\leq x\leq b}
2753:{\displaystyle f(x)\geq g(x)}
425:
363:. This is not true of other
294:fixed point of all isometries
185:In physics, if variations in
7480:, pp. 18, 189, 225–226)
5654:respectively. For any point
4852:
4830:
4808:
4786:
4764:
4742:
4720:
4695:
4673:
4648:
4558:
4537:
4433:The area of the triangle is
2589:with the horizontal line at
461:
458:
455:
277:The geometric centroid of a
166:, one often assumes uniform
7616:Advanced Euclidean Geometry
3671:then the centroid (denoted
7803:
7632:Holt, Rinehart and Winston
7614:Johnson, Roger A. (2007),
7605:Bourke, Paul (July 1997).
7589:(2nd ed.), New York:
7369:Protter & Morrey (1970
7357:Protter & Morrey (1970
7345:Protter & Morrey (1970
7216:Protter & Morrey (1970
7204:Protter & Morrey (1970
3401:
2505:with the vertical line at
2425:is the area of the figure
689:By geometric decomposition
400:
7756:Dynamic Geometry Sketches
7662:(2nd ed.), Reading:
7527:Kimberling, Clark (201).
7182:Pappus's centroid theorem
5647:{\displaystyle BC,CA,AB,}
5578:be its centroid, and let
1587:The centroid of a subset
482:Of a finite set of points
7652:Houghton Mifflin Company
3434:(the lines joining each
2225:{\displaystyle x_{k}=z.}
2169:{\displaystyle S_{k}(z)}
1470:should be the volume of
7247:The Mathematics Teacher
6832:three-dimensional space
3967:barycentric coordinates
1780:characteristic function
775:computing the centroid
189:are considered, then a
64:more precise citations.
7743:Centroid of a triangle
7626:Kay, David C. (1969),
7515:Altshiller-Court (1925
7466:Altshiller-Court (1925
7404:Altshiller-Court (1925
7330:Altshiller-Court (1925
7132:
7103:
7069:
7041:
7023:
6960:
6905:
6870:
6808:
6779:
6730:
6707:
6681:
6655:
6606:
6558:
6483:
6422:
6400:
6290:
6217:
6107:
6036:
5987:
5927:
5871:
5821:
5790:
5697:
5668:
5648:
5604:
5572:
5552:
5521:
5395:
5260:
5239:
5219:
5199:
5179:
5156:
5124:
5101:
5054:
4919:
4890:
4868:
4621:
4595:
4570:
4512:
4486:
4456:
4405:
4035:
4002:
3959:
3898:
3705:
3685:
3665:
3610:
3556:
3493:
3464:
3387:
3367:
3344:
3318:
3293:
3264:
3198:
3119:
3097:
2820:
2789:
2754:
2710:
2690:
2667:
2609:
2583:
2563:
2525:
2499:
2479:
2442:
2419:
2397:
2249:
2226:
2190:
2170:
2134:
2111:
2091:
2062:
1958:
1938:
1911:
1910:{\displaystyle g(x)=0}
1876:
1875:{\displaystyle x\in X}
1850:
1796:
1772:
1752:
1714:
1630:
1601:
1569:
1549:
1526:
1494:
1493:{\displaystyle X_{i},}
1464:
1434:
1221:
1201:
1178:
1158:
1138:
1118:
1091:
1064:
1063:{\displaystyle A_{i}.}
1034:
1009:
823:
796:
769:
707:
679:
594:
565:
500:
450:
441:
432:
372:translational symmetry
232:
195:can be defined as the
146:
97:
96:Centroid of a triangle
7133:
7104:
7070:
7042:
7003:
6961:
6906:
6871:
6818:Of a tetrahedron and
6809:
6780:
6731:
6708:
6682:
6656:
6607:
6559:
6457:
6423:
6401:
6264:
6218:
6081:
6037:
5988:
5928:
5872:
5822:
5791:
5698:
5669:
5649:
5605:
5603:{\displaystyle D,E,F}
5573:
5553:
5522:
5396:
5261:
5240:
5220:
5200:
5180:
5157:
5155:{\displaystyle A,B,C}
5125:
5102:
5055:
4920:
4891:
4869:
4622:
4596:
4578:In addition, for the
4571:
4513:
4487:
4457:
4406:
4036:
4034:{\displaystyle L,M,N}
4003:
4001:{\displaystyle a,b,c}
3974:trilinear coordinates
3960:
3899:
3706:
3686:
3666:
3611:
3557:
3501:Cartesian coordinates
3494:
3465:
3388:
3368:
3345:
3319:
3294:
3265:
3228:Of an L-shaped object
3199:
3120:
3098:
2821:
2790:
2755:
2711:
2691:
2668:
2610:
2584:
2564:
2526:
2500:
2480:
2443:
2420:
2398:
2250:
2227:
2191:
2171:
2135:
2112:
2092:
2090:{\displaystyle C_{k}}
2063:
1959:
1939:
1912:
1877:
1851:
1797:
1773:
1753:
1715:
1631:
1602:
1570:
1550:
1527:
1495:
1465:
1463:{\displaystyle A_{i}}
1435:
1222:
1202:
1179:
1159:
1139:
1119:
1117:{\displaystyle A_{i}}
1092:
1090:{\displaystyle A_{i}}
1070:Namely, the measures
1065:
1035:
1010:
824:
822:{\displaystyle A_{i}}
797:
795:{\displaystyle C_{i}}
770:
708:
680:
595:
566:
501:
449:
440:
431:
147:
95:
7711:"Geometric Centroid"
7694:on November 13, 2013
7259:10.5951/MT.53.1.0033
7113:
7087:
7053:
6976:
6919:
6895:
6869:{\displaystyle 3:1.}
6854:
6822:-dimensional simplex
6789:
6760:
6740:Of a cone or pyramid
6717:
6706:{\displaystyle i=0.}
6691:
6665:
6616:
6570:
6438:
6412:
6231:
6048:
5997:
5936:
5880:
5831:
5811:
5708:
5696:{\displaystyle ABC,}
5678:
5658:
5614:
5582:
5562:
5536:
5409:
5273:
5250:
5229:
5209:
5189:
5166:
5134:
5114:
5082:
4931:
4918:{\displaystyle ABC,}
4900:
4880:
4631:
4608:
4585:
4524:
4499:
4476:
4437:
4048:
4013:
3980:
3910:
3721:
3695:
3675:
3620:
3565:
3511:
3474:
3463:{\displaystyle 2:1,}
3445:
3377:
3354:
3328:
3305:
3277:
3248:
3129:
3109:
2832:
2798:
2764:
2720:
2700:
2680:
2675:continuous functions
2626:
2596:
2573:
2535:
2512:
2489:
2451:
2429:
2409:
2263:
2236:
2200:
2180:
2144:
2121:
2101:
2074:
1973:
1948:
1925:
1886:
1860:
1806:
1786:
1762:
1730:
1642:
1611:
1591:
1559:
1536:
1504:
1474:
1447:
1283:
1211:
1188:
1168:
1148:
1128:
1101:
1074:
1044:
1021:
1017:Holes in the figure
835:
806:
779:
717:
697:
604:
575:
510:
490:
170:, in which case the
136:
7533:Forum Geometricorum
7383:, pp. 458–460)
7102:{\displaystyle 3:5}
7068:{\displaystyle n+1}
6680:{\displaystyle i+1}
5558:be a triangle, let
5551:{\displaystyle ABC}
3343:{\displaystyle CD,}
3292:{\displaystyle CD.}
3263:{\displaystyle AB.}
3216:(a relative of the
3146:
2984:
2882:
2618:Of a bounded region
1583:By integral formula
263:Heron of Alexandria
216:geographical center
7708:Weisstein, Eric W.
7591:Barnes & Noble
7490:Bottomley, Henry.
7298:10.1007/BF03023072
7280:Knorr, W. (1978).
7128:
7126:
7099:
7065:
7037:
6970:, the centroid is
6956:
6901:
6866:
6804:
6802:
6775:
6773:
6744:The centroid of a
6729:{\displaystyle A,}
6726:
6703:
6677:
6651:
6602:
6554:
6418:
6396:
6213:
6032:
5983:
5923:
5867:
5817:
5786:
5693:
5664:
5644:
5600:
5568:
5548:
5517:
5391:
5256:
5235:
5215:
5195:
5178:{\displaystyle G.}
5175:
5152:
5120:
5097:
5095:
5064:isogonal conjugate
5050:
5022:
4915:
4886:
4864:
4862:
4620:{\displaystyle N,}
4617:
4591:
4566:
4511:{\displaystyle O,}
4508:
4482:
4452:
4450:
4401:
4399:
4031:
4008:and vertex angles
3998:
3955:
3953:
3938:
3923:
3894:
3840:
3783:
3740:
3701:
3681:
3661:
3606:
3552:
3489:
3487:
3460:
3426:The centroid of a
3383:
3366:{\displaystyle O.}
3363:
3340:
3317:{\displaystyle AB}
3314:
3289:
3260:
3194:
3132:
3115:
3093:
3091:
2995:
2970:
2868:
2816:
2785:
2750:
2706:
2686:
2663:
2608:{\displaystyle y.}
2605:
2579:
2559:
2524:{\displaystyle x,}
2521:
2495:
2475:
2441:{\displaystyle X,}
2438:
2415:
2393:
2248:{\displaystyle X.}
2245:
2222:
2186:
2166:
2133:{\displaystyle C,}
2130:
2107:
2087:
2058:
1954:
1937:{\displaystyle X.}
1934:
1907:
1872:
1846:
1792:
1768:
1748:
1710:
1626:
1597:
1565:
1548:{\displaystyle d,}
1545:
1532:for any dimension
1522:
1490:
1460:
1430:
1217:
1200:{\displaystyle X,}
1197:
1174:
1154:
1134:
1114:
1087:
1060:
1033:{\displaystyle X,}
1030:
1005:
988:
947:
904:
863:
819:
792:
765:
703:
675:
590:
561:
496:
451:
442:
433:
314:regular polyhedron
142:
98:
7777:Geometric centers
7517:, pp. 70–71)
7172:List of centroids
7125:
7001:
6904:{\displaystyle n}
6801:
6772:
6537:
6499:
6455:
6421:{\displaystyle A}
6379:
6341:
6262:
6196:
6158:
6079:
5820:{\displaystyle n}
5667:{\displaystyle P}
5571:{\displaystyle G}
5259:{\displaystyle G}
5238:{\displaystyle P}
5218:{\displaystyle G}
5198:{\displaystyle P}
5123:{\displaystyle P}
5094:
5030:
5021:
4992:
4966:
4940:
4889:{\displaystyle G}
4855:
4833:
4811:
4789:
4767:
4745:
4723:
4698:
4676:
4651:
4603:nine-point center
4594:{\displaystyle I}
4561:
4540:
4485:{\displaystyle H}
4449:
4099:
4086:
4073:
3952:
3937:
3922:
3839:
3782:
3739:
3713:triangle geometry
3704:{\displaystyle G}
3684:{\displaystyle C}
3486:
3424:
3423:
3386:{\displaystyle O}
3208:With an integraph
3118:{\displaystyle A}
2994:
2968:
2950:
2866:
2848:
2788:{\displaystyle ,}
2709:{\displaystyle g}
2689:{\displaystyle f}
2657:
2641:
2582:{\displaystyle X}
2498:{\displaystyle X}
2418:{\displaystyle A}
2388:
2377:
2324:
2313:
2189:{\displaystyle X}
2117:th coordinate of
2110:{\displaystyle k}
2053:
2045:
2019:
1957:{\displaystyle X}
1827:
1795:{\displaystyle X}
1771:{\displaystyle g}
1708:
1700:
1674:
1600:{\displaystyle X}
1568:{\displaystyle d}
1425:
1414:
1385:
1328:
1220:{\displaystyle 0}
1177:{\displaystyle p}
1157:{\displaystyle 1}
1137:{\displaystyle p}
1000:
979:
938:
916:
895:
854:
706:{\displaystyle X}
670:
499:{\displaystyle k}
469:along the shape.
466:
465:
408:Plumb line method
233:centre de gravité
192:center of gravity
145:{\displaystyle n}
90:
89:
82:
16:(Redirected from
7794:
7787:Triangle centers
7721:
7720:
7695:
7693:
7687:, archived from
7686:
7674:
7654:
7650:(6th ed.),
7642:
7628:College Geometry
7622:
7610:
7601:
7573:
7570:
7564:
7559:
7550:
7547:
7541:
7540:
7524:
7518:
7512:
7503:
7502:
7500:
7498:
7487:
7481:
7475:
7469:
7463:
7457:
7451:
7445:
7444:
7442:
7441:
7432:. Archived from
7425:
7419:
7413:
7407:
7401:
7395:
7390:
7384:
7378:
7372:
7366:
7360:
7354:
7348:
7342:
7333:
7327:
7318:
7317:
7277:
7271:
7270:
7242:
7236:
7225:
7219:
7213:
7207:
7201:
7167:-means algorithm
7166:
7149:Chebyshev center
7137:
7135:
7134:
7129:
7127:
7118:
7108:
7106:
7105:
7100:
7074:
7072:
7071:
7066:
7046:
7044:
7043:
7038:
7033:
7032:
7022:
7017:
7002:
7000:
6986:
6965:
6963:
6962:
6957:
6952:
6951:
6950:
6932:
6931:
6910:
6908:
6907:
6902:
6875:
6873:
6872:
6867:
6830:is an object in
6821:
6813:
6811:
6810:
6805:
6803:
6794:
6784:
6782:
6781:
6776:
6774:
6765:
6735:
6733:
6732:
6727:
6712:
6710:
6709:
6704:
6686:
6684:
6683:
6678:
6660:
6658:
6657:
6652:
6644:
6643:
6631:
6630:
6611:
6609:
6608:
6603:
6598:
6597:
6585:
6584:
6563:
6561:
6560:
6555:
6547:
6546:
6535:
6534:
6533:
6515:
6514:
6497:
6496:
6495:
6482:
6471:
6456:
6448:
6430:shoelace formula
6427:
6425:
6424:
6419:
6405:
6403:
6402:
6397:
6389:
6388:
6377:
6376:
6375:
6357:
6356:
6339:
6338:
6337:
6322:
6321:
6303:
6302:
6289:
6278:
6263:
6261:
6250:
6245:
6244:
6243:
6222:
6220:
6219:
6214:
6206:
6205:
6194:
6193:
6192:
6174:
6173:
6156:
6155:
6154:
6139:
6138:
6120:
6119:
6106:
6095:
6080:
6078:
6067:
6062:
6061:
6060:
6041:
6039:
6038:
6033:
6025:
6024:
6012:
6011:
5992:
5990:
5989:
5984:
5976:
5975:
5957:
5956:
5932:
5930:
5929:
5924:
5908:
5907:
5895:
5894:
5876:
5874:
5873:
5868:
5859:
5858:
5846:
5845:
5826:
5824:
5823:
5818:
5795:
5793:
5792:
5787:
5702:
5700:
5699:
5694:
5674:in the plane of
5673:
5671:
5670:
5665:
5653:
5651:
5650:
5645:
5609:
5607:
5606:
5601:
5577:
5575:
5574:
5569:
5557:
5555:
5554:
5549:
5526:
5524:
5523:
5518:
5510:
5509:
5494:
5493:
5478:
5477:
5456:
5455:
5440:
5439:
5424:
5423:
5400:
5398:
5397:
5392:
5387:
5386:
5368:
5367:
5352:
5351:
5336:
5335:
5320:
5319:
5304:
5303:
5288:
5287:
5267:
5265:
5263:
5262:
5257:
5244:
5242:
5241:
5236:
5224:
5222:
5221:
5216:
5204:
5202:
5201:
5196:
5184:
5182:
5181:
5176:
5161:
5159:
5158:
5153:
5129:
5127:
5126:
5121:
5106:
5104:
5103:
5098:
5096:
5087:
5059:
5057:
5056:
5051:
5031:
5028:
5023:
5014:
4993:
4990:
4967:
4964:
4941:
4938:
4924:
4922:
4921:
4916:
4895:
4893:
4892:
4887:
4873:
4871:
4870:
4865:
4863:
4856:
4851:
4843:
4834:
4829:
4821:
4812:
4807:
4799:
4790:
4785:
4777:
4768:
4763:
4755:
4746:
4741:
4733:
4724:
4719:
4711:
4699:
4694:
4686:
4677:
4672:
4664:
4652:
4647:
4639:
4626:
4624:
4623:
4618:
4600:
4598:
4597:
4592:
4575:
4573:
4572:
4567:
4562:
4557:
4549:
4541:
4536:
4528:
4517:
4515:
4514:
4509:
4491:
4489:
4488:
4483:
4461:
4459:
4458:
4453:
4451:
4442:
4410:
4408:
4407:
4402:
4400:
4282:
4167:
4100:
4092:
4087:
4079:
4074:
4066:
4042:
4040:
4038:
4037:
4032:
4007:
4005:
4004:
3999:
3964:
3962:
3961:
3956:
3954:
3945:
3939:
3930:
3924:
3915:
3903:
3901:
3900:
3895:
3890:
3889:
3880:
3879:
3867:
3866:
3854:
3853:
3841:
3832:
3823:
3822:
3810:
3809:
3797:
3796:
3784:
3775:
3772:
3771:
3741:
3732:
3710:
3708:
3707:
3702:
3690:
3688:
3687:
3682:
3670:
3668:
3667:
3662:
3654:
3653:
3641:
3640:
3615:
3613:
3612:
3607:
3599:
3598:
3586:
3585:
3561:
3559:
3558:
3553:
3545:
3544:
3532:
3531:
3498:
3496:
3495:
3490:
3488:
3479:
3469:
3467:
3466:
3461:
3420:
3415:
3408:
3392:
3390:
3389:
3384:
3372:
3370:
3369:
3364:
3349:
3347:
3346:
3341:
3323:
3321:
3320:
3315:
3298:
3296:
3295:
3290:
3269:
3267:
3266:
3261:
3239:
3203:
3201:
3200:
3195:
3187:
3186:
3153:
3152:
3145:
3140:
3124:
3122:
3121:
3116:
3102:
3100:
3099:
3094:
3092:
3078:
3077:
3044:
3043:
3037:
3036:
3003:
3002:
2996:
2987:
2983:
2978:
2969:
2961:
2952:
2951:
2943:
2926:
2925:
2892:
2891:
2881:
2876:
2867:
2859:
2850:
2849:
2841:
2825:
2823:
2822:
2817:
2794:
2792:
2791:
2786:
2760:on the interval
2759:
2757:
2756:
2751:
2715:
2713:
2712:
2707:
2695:
2693:
2692:
2687:
2672:
2670:
2669:
2664:
2659:
2658:
2650:
2643:
2642:
2634:
2614:
2612:
2611:
2606:
2588:
2586:
2585:
2580:
2568:
2566:
2565:
2560:
2549:
2548:
2547:
2530:
2528:
2527:
2522:
2504:
2502:
2501:
2496:
2484:
2482:
2481:
2476:
2465:
2464:
2463:
2447:
2445:
2444:
2439:
2424:
2422:
2421:
2416:
2402:
2400:
2399:
2394:
2389:
2384:
2375:
2365:
2364:
2363:
2346:
2341:
2340:
2339:
2325:
2320:
2311:
2301:
2300:
2299:
2282:
2277:
2276:
2275:
2254:
2252:
2251:
2246:
2231:
2229:
2228:
2223:
2212:
2211:
2195:
2193:
2192:
2187:
2175:
2173:
2172:
2167:
2156:
2155:
2139:
2137:
2136:
2131:
2116:
2114:
2113:
2108:
2096:
2094:
2093:
2088:
2086:
2085:
2067:
2065:
2064:
2059:
2054:
2052:
2043:
2026:
2017:
2007:
2006:
1990:
1985:
1984:
1963:
1961:
1960:
1955:
1943:
1941:
1940:
1935:
1916:
1914:
1913:
1908:
1881:
1879:
1878:
1873:
1855:
1853:
1852:
1847:
1825:
1820:
1819:
1814:
1801:
1799:
1798:
1793:
1777:
1775:
1774:
1769:
1757:
1755:
1754:
1749:
1744:
1743:
1738:
1719:
1717:
1716:
1711:
1709:
1707:
1698:
1681:
1672:
1652:
1635:
1633:
1632:
1627:
1625:
1624:
1619:
1606:
1604:
1603:
1598:
1574:
1572:
1571:
1566:
1554:
1552:
1551:
1546:
1531:
1529:
1528:
1523:
1518:
1517:
1512:
1499:
1497:
1496:
1491:
1486:
1485:
1469:
1467:
1466:
1461:
1459:
1458:
1439:
1437:
1436:
1431:
1426:
1423:
1415:
1413:
1412:
1411:
1396:
1395:
1386:
1378:
1373:
1372:
1362:
1361:
1360:
1339:
1338:
1329:
1321:
1310:
1309:
1293:
1265:
1253:
1241:
1226:
1224:
1223:
1218:
1206:
1204:
1203:
1198:
1183:
1181:
1180:
1175:
1163:
1161:
1160:
1155:
1143:
1141:
1140:
1135:
1123:
1121:
1120:
1115:
1113:
1112:
1096:
1094:
1093:
1088:
1086:
1085:
1069:
1067:
1066:
1061:
1056:
1055:
1039:
1037:
1036:
1031:
1014:
1012:
1011:
1006:
1001:
999:
998:
997:
987:
977:
976:
975:
966:
965:
960:
959:
958:
946:
936:
931:
930:
917:
915:
914:
913:
903:
893:
892:
891:
882:
881:
876:
875:
874:
862:
852:
847:
846:
828:
826:
825:
820:
818:
817:
801:
799:
798:
793:
791:
790:
774:
772:
771:
766:
761:
760:
742:
741:
729:
728:
712:
710:
709:
704:
684:
682:
681:
676:
671:
666:
665:
664:
659:
644:
643:
638:
629:
628:
623:
616:
611:
599:
597:
596:
591:
589:
588:
583:
570:
568:
567:
562:
560:
559:
554:
539:
538:
533:
524:
523:
518:
505:
503:
502:
497:
473:Balancing method
424:
235:
151:
149:
148:
143:
118:center of figure
114:geometric center
112:, also known as
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
7802:
7801:
7797:
7796:
7795:
7793:
7792:
7791:
7772:Affine geometry
7762:
7761:
7706:
7705:
7702:
7691:
7684:
7678:Sangwin, C.J.,
7677:
7657:
7645:
7625:
7613:
7604:
7584:
7581:
7576:
7571:
7567:
7560:
7553:
7548:
7544:
7526:
7525:
7521:
7513:
7506:
7496:
7494:
7489:
7488:
7484:
7476:
7472:
7464:
7460:
7452:
7448:
7439:
7437:
7428:
7426:
7422:
7414:
7410:
7402:
7398:
7391:
7387:
7379:
7375:
7367:
7363:
7355:
7351:
7343:
7336:
7328:
7321:
7279:
7278:
7274:
7244:
7243:
7239:
7226:
7222:
7214:
7210:
7202:
7195:
7191:
7186:
7164:
7144:
7111:
7110:
7085:
7084:
7081:
7079:Of a hemisphere
7051:
7050:
7024:
6990:
6974:
6973:
6942:
6923:
6917:
6916:
6893:
6892:
6888:of a triangle.
6852:
6851:
6824:
6819:
6787:
6786:
6758:
6757:
6742:
6715:
6714:
6689:
6688:
6663:
6662:
6635:
6622:
6614:
6613:
6589:
6576:
6568:
6567:
6538:
6519:
6500:
6487:
6436:
6435:
6410:
6409:
6380:
6361:
6342:
6329:
6307:
6294:
6254:
6234:
6229:
6228:
6197:
6178:
6159:
6146:
6124:
6111:
6071:
6051:
6046:
6045:
6016:
6003:
5995:
5994:
5961:
5942:
5934:
5933:
5899:
5886:
5878:
5877:
5850:
5837:
5829:
5828:
5809:
5808:
5801:
5706:
5705:
5676:
5675:
5656:
5655:
5612:
5611:
5580:
5579:
5560:
5559:
5534:
5533:
5501:
5485:
5469:
5447:
5431:
5415:
5407:
5406:
5378:
5359:
5343:
5327:
5311:
5295:
5279:
5271:
5270:
5248:
5247:
5246:
5227:
5226:
5207:
5206:
5187:
5186:
5164:
5163:
5132:
5131:
5112:
5111:
5080:
5079:
5068:symmedian point
4929:
4928:
4898:
4897:
4878:
4877:
4861:
4860:
4844:
4835:
4822:
4817:
4816:
4800:
4791:
4778:
4773:
4772:
4756:
4747:
4734:
4729:
4728:
4712:
4700:
4687:
4682:
4681:
4665:
4653:
4640:
4629:
4628:
4606:
4605:
4583:
4582:
4550:
4529:
4522:
4521:
4497:
4496:
4474:
4473:
4435:
4434:
4428:medial triangle
4398:
4397:
4280:
4279:
4165:
4164:
4058:
4046:
4045:
4011:
4010:
4009:
3978:
3977:
3908:
3907:
3871:
3858:
3845:
3814:
3801:
3788:
3719:
3718:
3693:
3692:
3673:
3672:
3645:
3632:
3618:
3617:
3590:
3577:
3563:
3562:
3536:
3523:
3509:
3508:
3472:
3471:
3443:
3442:
3406:
3404:Triangle center
3400:
3375:
3374:
3352:
3351:
3326:
3325:
3324:and also along
3303:
3302:
3275:
3274:
3246:
3245:
3230:
3222:Green's theorem
3210:
3127:
3126:
3107:
3106:
3090:
3089:
2953:
2938:
2937:
2851:
2830:
2829:
2796:
2795:
2762:
2761:
2718:
2717:
2698:
2697:
2678:
2677:
2624:
2623:
2620:
2594:
2593:
2571:
2570:
2538:
2533:
2532:
2510:
2509:
2487:
2486:
2454:
2449:
2448:
2427:
2426:
2407:
2406:
2354:
2347:
2330:
2290:
2283:
2266:
2261:
2260:
2234:
2233:
2203:
2198:
2197:
2178:
2177:
2147:
2142:
2141:
2119:
2118:
2099:
2098:
2077:
2072:
2071:
2027:
1998:
1991:
1976:
1971:
1970:
1946:
1945:
1923:
1922:
1884:
1883:
1858:
1857:
1809:
1804:
1803:
1784:
1783:
1760:
1759:
1733:
1728:
1727:
1682:
1653:
1640:
1639:
1614:
1609:
1608:
1589:
1588:
1585:
1557:
1556:
1534:
1533:
1507:
1502:
1501:
1477:
1472:
1471:
1450:
1445:
1444:
1403:
1387:
1364:
1363:
1352:
1330:
1301:
1294:
1281:
1280:
1273:
1272:
1271:
1270:
1269:
1266:
1258:
1257:
1254:
1246:
1245:
1242:
1209:
1208:
1186:
1185:
1166:
1165:
1146:
1145:
1126:
1125:
1104:
1099:
1098:
1077:
1072:
1071:
1047:
1042:
1041:
1019:
1018:
989:
978:
967:
950:
948:
937:
922:
905:
894:
883:
866:
864:
853:
838:
833:
832:
809:
804:
803:
782:
777:
776:
752:
733:
720:
715:
714:
695:
694:
691:
654:
633:
618:
617:
602:
601:
578:
573:
572:
549:
528:
513:
508:
507:
488:
487:
484:
475:
410:
405:
399:
380:
310:regular polygon
275:
224:
205:specific weight
157:Euclidean space
134:
133:
130:arithmetic mean
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
7800:
7798:
7790:
7789:
7784:
7779:
7774:
7764:
7763:
7760:
7759:
7749:
7739:
7730:
7722:
7701:
7700:External links
7698:
7697:
7696:
7675:
7664:Addison-Wesley
7655:
7643:
7623:
7611:
7602:
7580:
7577:
7575:
7574:
7565:
7551:
7542:
7519:
7504:
7482:
7470:
7468:, p. 101)
7458:
7456:, p. 173)
7446:
7420:
7418:, p. 184)
7408:
7396:
7385:
7373:
7371:, p. 528)
7361:
7359:, p. 527)
7349:
7347:, p. 526)
7334:
7319:
7292:(2): 102–109.
7272:
7237:
7220:
7218:, p. 521)
7208:
7206:, p. 520)
7192:
7190:
7187:
7185:
7184:
7179:
7174:
7169:
7161:
7156:
7151:
7145:
7143:
7140:
7124:
7121:
7109:(i.e. it lies
7098:
7095:
7092:
7080:
7077:
7075:equal masses.
7064:
7061:
7058:
7036:
7031:
7027:
7021:
7016:
7013:
7010:
7006:
6999:
6996:
6993:
6989:
6984:
6981:
6955:
6949:
6945:
6941:
6938:
6935:
6930:
6926:
6900:
6865:
6862:
6859:
6823:
6816:
6800:
6797:
6771:
6768:
6741:
6738:
6725:
6722:
6702:
6699:
6696:
6676:
6673:
6670:
6650:
6647:
6642:
6638:
6634:
6629:
6625:
6621:
6601:
6596:
6592:
6588:
6583:
6579:
6575:
6553:
6550:
6545:
6541:
6532:
6529:
6526:
6522:
6518:
6513:
6510:
6507:
6503:
6494:
6490:
6486:
6481:
6478:
6475:
6470:
6467:
6464:
6460:
6454:
6451:
6446:
6443:
6417:
6395:
6392:
6387:
6383:
6374:
6371:
6368:
6364:
6360:
6355:
6352:
6349:
6345:
6336:
6332:
6328:
6325:
6320:
6317:
6314:
6310:
6306:
6301:
6297:
6293:
6288:
6285:
6282:
6277:
6274:
6271:
6267:
6260:
6257:
6253:
6248:
6242:
6237:
6212:
6209:
6204:
6200:
6191:
6188:
6185:
6181:
6177:
6172:
6169:
6166:
6162:
6153:
6149:
6145:
6142:
6137:
6134:
6131:
6127:
6123:
6118:
6114:
6110:
6105:
6102:
6099:
6094:
6091:
6088:
6084:
6077:
6074:
6070:
6065:
6059:
6054:
6031:
6028:
6023:
6019:
6015:
6010:
6006:
6002:
5982:
5979:
5974:
5971:
5968:
5964:
5960:
5955:
5952:
5949:
5945:
5941:
5921:
5918:
5914:
5911:
5906:
5902:
5898:
5893:
5889:
5885:
5865:
5862:
5857:
5853:
5849:
5844:
5840:
5836:
5816:
5800:
5797:
5785:
5782:
5779:
5776:
5773:
5770:
5767:
5764:
5761:
5758:
5755:
5752:
5749:
5746:
5743:
5740:
5737:
5734:
5731:
5728:
5725:
5722:
5719:
5716:
5713:
5692:
5689:
5686:
5683:
5663:
5643:
5640:
5637:
5634:
5631:
5628:
5625:
5622:
5619:
5599:
5596:
5593:
5590:
5587:
5567:
5547:
5544:
5541:
5516:
5513:
5508:
5504:
5500:
5497:
5492:
5488:
5484:
5481:
5476:
5472:
5468:
5465:
5462:
5459:
5454:
5450:
5446:
5443:
5438:
5434:
5430:
5427:
5422:
5418:
5414:
5390:
5385:
5381:
5377:
5374:
5371:
5366:
5362:
5358:
5355:
5350:
5346:
5342:
5339:
5334:
5330:
5326:
5323:
5318:
5314:
5310:
5307:
5302:
5298:
5294:
5291:
5286:
5282:
5278:
5255:
5234:
5214:
5194:
5174:
5171:
5151:
5148:
5145:
5142:
5139:
5119:
5093:
5090:
5049:
5046:
5043:
5040:
5037:
5034:
5026:
5020:
5017:
5011:
5008:
5005:
5002:
4999:
4996:
4988:
4985:
4982:
4979:
4976:
4973:
4970:
4962:
4959:
4956:
4953:
4950:
4947:
4944:
4936:
4914:
4911:
4908:
4905:
4885:
4859:
4854:
4850:
4847:
4841:
4838:
4836:
4832:
4828:
4825:
4819:
4818:
4815:
4810:
4806:
4803:
4797:
4794:
4792:
4788:
4784:
4781:
4775:
4774:
4771:
4766:
4762:
4759:
4753:
4750:
4748:
4744:
4740:
4737:
4731:
4730:
4727:
4722:
4718:
4715:
4709:
4706:
4703:
4701:
4697:
4693:
4690:
4684:
4683:
4680:
4675:
4671:
4668:
4662:
4659:
4656:
4654:
4650:
4646:
4643:
4637:
4636:
4616:
4613:
4590:
4565:
4560:
4556:
4553:
4547:
4544:
4539:
4535:
4532:
4507:
4504:
4481:
4448:
4445:
4420:Spieker center
4416:linear density
4396:
4393:
4390:
4387:
4384:
4381:
4378:
4375:
4372:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4348:
4345:
4342:
4339:
4336:
4333:
4330:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4291:
4288:
4285:
4283:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4254:
4251:
4248:
4245:
4242:
4239:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4168:
4166:
4163:
4160:
4157:
4154:
4151:
4148:
4145:
4142:
4139:
4136:
4133:
4130:
4127:
4124:
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4098:
4095:
4090:
4085:
4082:
4077:
4072:
4069:
4064:
4061:
4059:
4057:
4054:
4053:
4030:
4027:
4024:
4021:
4018:
3997:
3994:
3991:
3988:
3985:
3951:
3948:
3942:
3936:
3933:
3927:
3921:
3918:
3893:
3888:
3883:
3878:
3874:
3870:
3865:
3861:
3857:
3852:
3848:
3844:
3838:
3835:
3829:
3826:
3821:
3817:
3813:
3808:
3804:
3800:
3795:
3791:
3787:
3781:
3778:
3770:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3738:
3735:
3729:
3726:
3700:
3680:
3660:
3657:
3652:
3648:
3644:
3639:
3635:
3631:
3628:
3625:
3605:
3602:
3597:
3593:
3589:
3584:
3580:
3576:
3573:
3570:
3551:
3548:
3543:
3539:
3535:
3530:
3526:
3522:
3519:
3516:
3485:
3482:
3459:
3456:
3453:
3450:
3422:
3421:
3402:Main article:
3399:
3396:
3395:
3394:
3382:
3362:
3359:
3339:
3336:
3333:
3313:
3310:
3299:
3288:
3285:
3282:
3270:
3259:
3256:
3253:
3229:
3226:
3209:
3206:
3193:
3190:
3185:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3151:
3144:
3139:
3135:
3114:
3088:
3085:
3082:
3076:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3050:
3047:
3042:
3035:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3009:
3006:
3001:
2993:
2990:
2982:
2977:
2973:
2967:
2964:
2959:
2956:
2954:
2949:
2946:
2940:
2939:
2936:
2933:
2930:
2924:
2919:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2890:
2885:
2880:
2875:
2871:
2865:
2862:
2857:
2854:
2852:
2847:
2844:
2838:
2837:
2815:
2812:
2809:
2806:
2803:
2784:
2781:
2778:
2775:
2772:
2769:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2728:
2725:
2705:
2685:
2662:
2656:
2653:
2646:
2640:
2637:
2631:
2619:
2616:
2604:
2601:
2578:
2558:
2555:
2552:
2546:
2541:
2520:
2517:
2494:
2474:
2471:
2468:
2462:
2457:
2437:
2434:
2414:
2392:
2387:
2383:
2380:
2374:
2371:
2368:
2362:
2357:
2353:
2350:
2344:
2338:
2333:
2328:
2323:
2319:
2316:
2310:
2307:
2304:
2298:
2293:
2289:
2286:
2280:
2274:
2269:
2244:
2241:
2221:
2218:
2215:
2210:
2206:
2185:
2165:
2162:
2159:
2154:
2150:
2129:
2126:
2106:
2084:
2080:
2057:
2051:
2048:
2042:
2039:
2036:
2033:
2030:
2025:
2022:
2016:
2013:
2010:
2005:
2001:
1997:
1994:
1988:
1983:
1979:
1953:
1933:
1930:
1906:
1903:
1900:
1897:
1894:
1891:
1871:
1868:
1865:
1845:
1842:
1839:
1836:
1833:
1830:
1824:
1818:
1813:
1791:
1782:of the subset
1767:
1747:
1742:
1737:
1706:
1703:
1697:
1694:
1691:
1688:
1685:
1680:
1677:
1671:
1668:
1665:
1662:
1659:
1656:
1650:
1647:
1623:
1618:
1596:
1584:
1581:
1579:of the parts.
1564:
1544:
1541:
1521:
1516:
1511:
1489:
1484:
1480:
1457:
1453:
1429:
1421:
1418:
1410:
1406:
1402:
1399:
1394:
1390:
1384:
1381:
1376:
1371:
1367:
1359:
1355:
1351:
1348:
1345:
1342:
1337:
1333:
1327:
1324:
1319:
1316:
1313:
1308:
1304:
1300:
1297:
1291:
1288:
1267:
1260:
1259:
1255:
1248:
1247:
1243:
1236:
1235:
1234:
1233:
1232:
1216:
1196:
1193:
1173:
1153:
1133:
1111:
1107:
1084:
1080:
1059:
1054:
1050:
1029:
1026:
1004:
996:
992:
986:
982:
974:
970:
964:
957:
953:
945:
941:
934:
929:
925:
920:
912:
908:
902:
898:
890:
886:
880:
873:
869:
861:
857:
850:
845:
841:
816:
812:
789:
785:
764:
759:
755:
751:
748:
745:
740:
736:
732:
727:
723:
702:
690:
687:
674:
669:
663:
658:
653:
650:
647:
642:
637:
632:
627:
622:
614:
610:
587:
582:
558:
553:
548:
545:
542:
537:
532:
527:
522:
517:
495:
483:
480:
474:
471:
464:
463:
460:
457:
453:
452:
443:
434:
409:
406:
398:
397:Determination
395:
379:
376:
365:quadrilaterals
350:superellipsoid
298:symmetry group
274:
271:
228:center of mass
223:
220:
199:of all points
179:center of mass
141:
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7799:
7788:
7785:
7783:
7780:
7778:
7775:
7773:
7770:
7769:
7767:
7757:
7753:
7750:
7748:
7744:
7740:
7738:
7734:
7731:
7728:
7727:
7723:
7718:
7717:
7712:
7709:
7704:
7703:
7699:
7690:
7683:
7682:
7676:
7673:
7669:
7665:
7661:
7656:
7653:
7649:
7644:
7641:
7637:
7633:
7629:
7624:
7621:
7617:
7612:
7608:
7603:
7600:
7596:
7592:
7588:
7583:
7582:
7578:
7569:
7566:
7563:
7562:Bourke (1997)
7558:
7556:
7552:
7546:
7543:
7538:
7534:
7530:
7523:
7520:
7516:
7511:
7509:
7505:
7493:
7486:
7483:
7479:
7474:
7471:
7467:
7462:
7459:
7455:
7454:Johnson (2007
7450:
7447:
7436:on 2012-04-19
7435:
7431:
7424:
7421:
7417:
7412:
7409:
7406:, p. 65)
7405:
7400:
7397:
7394:
7389:
7386:
7382:
7377:
7374:
7370:
7365:
7362:
7358:
7353:
7350:
7346:
7341:
7339:
7335:
7332:, p. 66)
7331:
7326:
7324:
7320:
7315:
7311:
7307:
7303:
7299:
7295:
7291:
7287:
7283:
7276:
7273:
7268:
7264:
7260:
7256:
7252:
7248:
7241:
7238:
7235:
7231:
7230:
7224:
7221:
7217:
7212:
7209:
7205:
7200:
7198:
7194:
7188:
7183:
7180:
7178:
7175:
7173:
7170:
7168:
7162:
7160:
7157:
7155:
7154:Circular mean
7152:
7150:
7147:
7146:
7141:
7139:
7122:
7119:
7096:
7093:
7090:
7078:
7076:
7062:
7059:
7056:
7047:
7034:
7029:
7025:
7019:
7014:
7011:
7008:
7004:
6997:
6994:
6991:
6987:
6982:
6979:
6971:
6969:
6953:
6947:
6943:
6939:
6936:
6933:
6928:
6924:
6914:
6911:-dimensional
6898:
6889:
6887:
6883:
6879:
6863:
6860:
6857:
6849:
6845:
6841:
6837:
6833:
6829:
6817:
6815:
6798:
6795:
6769:
6766:
6755:
6751:
6747:
6739:
6737:
6723:
6720:
6700:
6697:
6694:
6674:
6671:
6668:
6648:
6640:
6636:
6632:
6627:
6623:
6594:
6590:
6586:
6581:
6577:
6564:
6551:
6543:
6539:
6530:
6527:
6524:
6520:
6516:
6511:
6508:
6505:
6501:
6492:
6488:
6479:
6476:
6473:
6468:
6465:
6462:
6458:
6452:
6449:
6444:
6441:
6433:
6431:
6415:
6406:
6393:
6385:
6381:
6372:
6369:
6366:
6362:
6358:
6353:
6350:
6347:
6343:
6334:
6330:
6318:
6315:
6312:
6308:
6304:
6299:
6295:
6286:
6283:
6280:
6275:
6272:
6269:
6265:
6258:
6255:
6251:
6246:
6235:
6226:
6223:
6210:
6202:
6198:
6189:
6186:
6183:
6179:
6175:
6170:
6167:
6164:
6160:
6151:
6147:
6135:
6132:
6129:
6125:
6121:
6116:
6112:
6103:
6100:
6097:
6092:
6089:
6086:
6082:
6075:
6072:
6068:
6063:
6052:
6043:
6029:
6021:
6017:
6013:
6008:
6004:
5993:is the point
5980:
5972:
5969:
5966:
5962:
5958:
5953:
5950:
5947:
5943:
5919:
5916:
5912:
5904:
5900:
5896:
5891:
5887:
5863:
5855:
5851:
5847:
5842:
5838:
5814:
5806:
5798:
5796:
5783:
5780:
5777:
5774:
5771:
5765:
5762:
5759:
5756:
5753:
5750:
5747:
5744:
5738:
5735:
5732:
5729:
5726:
5723:
5720:
5717:
5714:
5711:
5703:
5690:
5687:
5684:
5681:
5661:
5641:
5638:
5635:
5632:
5629:
5626:
5623:
5620:
5617:
5597:
5594:
5591:
5588:
5585:
5565:
5545:
5542:
5539:
5530:
5527:
5514:
5506:
5502:
5498:
5495:
5490:
5486:
5482:
5479:
5474:
5470:
5466:
5460:
5457:
5452:
5448:
5444:
5441:
5436:
5432:
5428:
5425:
5420:
5416:
5412:
5404:
5401:
5388:
5383:
5379:
5375:
5372:
5369:
5364:
5360:
5356:
5353:
5348:
5344:
5340:
5337:
5332:
5328:
5324:
5321:
5316:
5312:
5308:
5305:
5300:
5296:
5292:
5289:
5284:
5280:
5276:
5268:
5253:
5232:
5212:
5192:
5172:
5169:
5162:and centroid
5149:
5146:
5143:
5140:
5137:
5117:
5108:
5091:
5088:
5077:
5071:
5069:
5065:
5060:
5047:
5041:
5038:
5035:
5029:Area of
5018:
5015:
5009:
5003:
5000:
4997:
4991:Area of
4983:
4977:
4974:
4971:
4965:Area of
4957:
4951:
4948:
4945:
4939:Area of
4926:
4912:
4909:
4906:
4903:
4883:
4874:
4857:
4848:
4845:
4839:
4837:
4826:
4823:
4813:
4804:
4801:
4795:
4793:
4782:
4779:
4769:
4760:
4757:
4751:
4749:
4738:
4735:
4725:
4716:
4713:
4707:
4704:
4702:
4691:
4688:
4678:
4669:
4666:
4660:
4657:
4655:
4644:
4641:
4614:
4611:
4604:
4588:
4581:
4576:
4563:
4554:
4551:
4545:
4542:
4533:
4530:
4519:
4505:
4502:
4495:
4479:
4472:
4468:
4463:
4446:
4443:
4431:
4429:
4425:
4421:
4417:
4411:
4394:
4391:
4388:
4385:
4382:
4379:
4376:
4373:
4370:
4367:
4364:
4361:
4358:
4355:
4352:
4349:
4346:
4343:
4340:
4337:
4334:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4304:
4301:
4298:
4295:
4292:
4289:
4286:
4284:
4276:
4273:
4270:
4267:
4264:
4261:
4258:
4255:
4252:
4249:
4246:
4243:
4240:
4237:
4234:
4231:
4228:
4225:
4222:
4219:
4216:
4213:
4210:
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4204:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4174:
4171:
4169:
4161:
4158:
4155:
4152:
4149:
4146:
4143:
4140:
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4134:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4101:
4096:
4093:
4088:
4083:
4080:
4075:
4070:
4067:
4062:
4060:
4055:
4043:
4028:
4025:
4022:
4019:
4016:
3995:
3992:
3989:
3986:
3983:
3975:
3970:
3968:
3949:
3946:
3940:
3934:
3931:
3925:
3919:
3916:
3904:
3891:
3876:
3872:
3868:
3863:
3859:
3855:
3850:
3846:
3836:
3833:
3827:
3819:
3815:
3811:
3806:
3802:
3798:
3793:
3789:
3779:
3776:
3763:
3757:
3754:
3751:
3748:
3745:
3736:
3733:
3727:
3724:
3716:
3714:
3698:
3678:
3658:
3650:
3646:
3642:
3637:
3633:
3626:
3623:
3603:
3595:
3591:
3587:
3582:
3578:
3571:
3568:
3549:
3541:
3537:
3533:
3528:
3524:
3517:
3514:
3506:
3502:
3483:
3480:
3457:
3454:
3451:
3448:
3441:
3437:
3433:
3429:
3419:
3414:
3410:
3409:
3405:
3398:Of a triangle
3397:
3380:
3360:
3357:
3337:
3334:
3331:
3311:
3308:
3300:
3286:
3283:
3280:
3271:
3257:
3254:
3251:
3242:
3241:
3240:
3238:
3233:
3227:
3225:
3223:
3219:
3215:
3207:
3205:
3191:
3188:
3175:
3169:
3166:
3160:
3154:
3142:
3137:
3133:
3112:
3103:
3086:
3083:
3080:
3066:
3060:
3057:
3051:
3045:
3025:
3019:
3016:
3010:
3004:
2991:
2988:
2980:
2975:
2971:
2965:
2962:
2957:
2955:
2944:
2934:
2931:
2928:
2914:
2908:
2905:
2899:
2893:
2883:
2878:
2873:
2869:
2863:
2860:
2855:
2853:
2842:
2827:
2813:
2810:
2807:
2804:
2801:
2782:
2776:
2773:
2770:
2744:
2738:
2735:
2729:
2723:
2703:
2683:
2676:
2651:
2644:
2635:
2622:The centroid
2617:
2615:
2602:
2599:
2592:
2576:
2553:
2539:
2518:
2515:
2508:
2492:
2469:
2455:
2435:
2432:
2412:
2403:
2390:
2385:
2381:
2378:
2369:
2355:
2351:
2348:
2342:
2331:
2326:
2321:
2317:
2314:
2305:
2291:
2287:
2284:
2278:
2267:
2258:
2255:
2242:
2239:
2219:
2216:
2213:
2208:
2204:
2183:
2160:
2152:
2148:
2127:
2124:
2104:
2082:
2078:
2068:
2055:
2049:
2046:
2037:
2031:
2028:
2023:
2020:
2011:
2003:
1999:
1995:
1992:
1986:
1981:
1977:
1968:
1965:
1951:
1931:
1928:
1920:
1904:
1901:
1895:
1889:
1869:
1866:
1863:
1843:
1840:
1834:
1828:
1822:
1816:
1789:
1781:
1765:
1745:
1740:
1725:
1720:
1704:
1701:
1692:
1686:
1683:
1678:
1675:
1666:
1660:
1657:
1654:
1648:
1645:
1637:
1621:
1594:
1582:
1580:
1578:
1575:-dimensional
1562:
1542:
1539:
1519:
1514:
1487:
1482:
1478:
1455:
1451:
1441:
1427:
1419:
1416:
1408:
1404:
1400:
1397:
1392:
1388:
1382:
1379:
1374:
1369:
1365:
1357:
1353:
1349:
1346:
1343:
1340:
1335:
1331:
1325:
1322:
1317:
1314:
1311:
1306:
1302:
1298:
1295:
1289:
1286:
1278:
1264:
1252:
1244:(a) 2D Object
1240:
1231:
1228:
1214:
1194:
1191:
1171:
1151:
1131:
1109:
1105:
1082:
1078:
1057:
1052:
1048:
1027:
1024:
1015:
1002:
994:
990:
984:
980:
972:
968:
962:
955:
951:
943:
939:
932:
927:
923:
918:
910:
906:
900:
896:
888:
884:
878:
871:
867:
859:
855:
848:
843:
839:
830:
814:
810:
787:
783:
762:
757:
753:
749:
746:
743:
738:
734:
730:
725:
721:
700:
688:
686:
672:
667:
661:
651:
648:
645:
640:
630:
625:
612:
585:
556:
546:
543:
540:
535:
525:
520:
493:
481:
479:
472:
470:
454:
448:
444:
439:
435:
430:
426:
422:
419:
415:
414:planar lamina
407:
404:
396:
394:
392:
387:
385:
377:
375:
373:
368:
366:
362:
358:
357:parallelogram
353:
351:
347:
343:
339:
335:
331:
327:
323:
319:
315:
311:
307:
303:
299:
295:
290:
288:
284:
280:
272:
270:
268:
264:
260:
259:
254:
249:
247:
243:
237:
234:
229:
221:
219:
217:
213:
208:
206:
202:
198:
197:weighted mean
194:
193:
188:
183:
181:
180:
175:
174:
169:
165:
160:
158:
155:
139:
131:
127:
123:
119:
115:
111:
107:
103:
94:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
7737:cut-the-knot
7725:
7714:
7689:the original
7680:
7659:
7647:
7630:, New York:
7627:
7615:
7586:
7568:
7545:
7536:
7532:
7522:
7497:27 September
7495:. Retrieved
7485:
7473:
7461:
7449:
7438:. Retrieved
7434:the original
7423:
7411:
7399:
7388:
7381:Larson (1998
7376:
7364:
7352:
7289:
7285:
7275:
7253:(1): 33–35.
7250:
7246:
7240:
7234:Google Books
7227:
7223:
7211:
7159:Fréchet mean
7082:
7048:
6972:
6890:
6881:
6847:
6843:
6839:
6825:
6743:
6565:
6434:
6407:
6227:
6224:
6044:
5802:
5799:Of a polygon
5704:
5531:
5528:
5405:
5402:
5269:
5109:
5072:
5061:
4927:
4875:
4577:
4520:
4494:circumcenter
4469:between its
4464:
4432:
4412:
4044:
3971:
3905:
3717:
3425:
3234:
3231:
3211:
3104:
2828:
2826:is given by
2621:
2404:
2259:
2256:
2069:
1969:
1966:
1721:
1638:
1586:
1442:
1274:
1229:
1016:
831:
692:
485:
476:
467:
411:
388:
381:
369:
354:
346:superellipse
291:
276:
266:
256:
250:
238:
225:
209:
190:
184:
177:
171:
168:mass density
161:
126:solid figure
122:plane figure
117:
113:
109:
99:
76:
67:
48:
6878:Monge point
6828:tetrahedron
5807:defined by
4471:orthocenter
1921:of the set
1424: units
1227:otherwise.
1184:belongs to
302:hyperplanes
154:dimensional
102:mathematics
62:introducing
7766:Categories
7579:References
7539:: 135–139.
7440:2012-06-02
6886:Euler line
6882:Euler line
6408:and where
4467:Euler line
3373:The point
3218:planimeter
2716:such that
1722:where the
401:See also:
273:Properties
246:Archimedes
173:barycenter
70:April 2013
45:references
18:Medicenter
7716:MathWorld
7478:Kay (1969
7416:Kay (1969
7314:122021219
7306:0343-6993
7005:∑
6937:…
6517:−
6477:−
6459:∑
6359:−
6284:−
6266:∑
6176:−
6101:−
6083:∑
5970:−
5951:−
5917:…
5827:vertices
5736:≤
5076:trapezoid
5033:△
4995:△
4969:△
4943:△
4853:¯
4831:¯
4809:¯
4787:¯
4765:¯
4743:¯
4721:¯
4696:¯
4674:¯
4649:¯
4559:¯
4538:¯
4389:
4383:⋅
4377:
4365:
4353:
4347:⋅
4341:
4329:
4317:
4311:⋅
4305:
4293:
4274:
4268:⋅
4262:
4250:
4238:
4232:⋅
4226:
4214:
4202:
4196:⋅
4190:
4178:
4159:
4147:
4135:
3214:integraph
3167:−
3134:∫
3058:−
2972:∫
2948:¯
2906:−
2870:∫
2846:¯
2811:≤
2805:≤
2736:≥
2655:¯
2639:¯
2349:∫
2285:∫
2029:∫
1993:∫
1867:∈
1724:integrals
1684:∫
1655:∫
1417:≈
1401:π
1398:−
1350:π
1347:×
1341:−
1318:×
1299:×
981:∑
940:∑
897:∑
856:∑
802:and area
747:…
649:⋯
544:…
418:plumbline
361:diagonals
342:ellipsoid
322:rectangle
267:Mechanics
212:geography
203:by their
7672:76087042
7640:69012075
7599:52013504
7267:27956057
7142:See also
6848:centroid
6844:bimedian
6661:meaning
4627:we have
4580:incenter
4492:and its
4424:incenter
3503:are the
3428:triangle
2591:ordinate
2507:abscissa
1577:measures
378:Examples
318:cylinder
306:symmetry
258:Elements
244:credits
201:weighted
164:geometry
110:centroid
7393:Sangwin
6968:vectors
6913:simplex
6750:pyramid
5805:polygon
4426:of the
3432:medians
2097:is the
1919:measure
1778:is the
506:points
384:medians
338:ellipse
326:rhombus
296:in its
222:History
187:gravity
128:is the
120:, of a
106:physics
58:improve
7670:
7638:
7597:
7312:
7304:
7265:
7177:Medoid
6840:median
6536:
6498:
6378:
6340:
6195:
6157:
6042:where
3436:vertex
3105:where
2405:where
2376:
2312:
2070:where
2044:
2018:
1826:
1699:
1673:
334:sphere
330:circle
279:convex
253:Euclid
242:Bossut
108:, the
47:, but
7782:Means
7692:(PDF)
7685:(PDF)
7620:Dover
7310:S2CID
7263:JSTOR
7189:Notes
6836:faces
4925:then
4422:(the
3715:) is
3505:means
3440:ratio
1315:13.33
391:below
285:or a
7745:and
7668:LCCN
7636:LCCN
7595:LCCN
7499:2013
7302:ISSN
6754:apex
6746:cone
6225:and
5532:Let
5245:and
5110:Let
5062:The
4840:<
4796:<
4752:<
4601:and
3616:and
2696:and
2531:and
2140:and
1882:and
1758:and
1207:and
462:(c)
459:(b)
456:(a)
287:bowl
283:ring
104:and
7754:at
7735:at
7294:doi
7255:doi
7232:at
6748:or
4876:If
4386:sec
4374:sec
4362:sec
4350:sec
4338:sec
4326:sec
4314:sec
4302:sec
4290:sec
4271:cos
4259:cos
4247:cos
4235:cos
4223:cos
4211:cos
4199:cos
4187:cos
4175:cos
4156:csc
4144:csc
4132:csc
3972:In
3965:in
3711:in
3212:An
3204:).
1856:if
1802:of
1607:of
1420:8.5
1405:2.5
1354:2.5
1164:if
1144:is
600:is
571:in
304:of
210:In
176:or
162:In
124:or
116:or
100:In
7768::
7713:.
7666:,
7634:,
7618:,
7593:,
7554:^
7537:10
7535:.
7531:.
7507:^
7337:^
7322:^
7308:.
7300:.
7288:.
7284:.
7261:.
7251:53
7249:.
7196:^
6864:1.
6826:A
6701:0.
6432::
5074:a
5070:.
3969:.
3224:.
1389:10
1366:10
1332:10
1303:10
393:.
367:.
348:,
344:,
340:,
336:,
332:,
328:,
324:,
320:,
316:,
312:,
218:.
207:.
159:.
7719:.
7609:.
7501:.
7443:.
7316:.
7296::
7290:1
7269:.
7257::
7165:k
7123:8
7120:3
7097:5
7094::
7091:3
7063:1
7060:+
7057:n
7035:.
7030:i
7026:v
7020:n
7015:0
7012:=
7009:i
6998:1
6995:+
6992:n
6988:1
6983:=
6980:C
6954:,
6948:n
6944:v
6940:,
6934:,
6929:0
6925:v
6899:n
6861::
6858:3
6820:n
6799:3
6796:1
6770:4
6767:1
6724:,
6721:A
6698:=
6695:i
6675:1
6672:+
6669:i
6649:,
6646:)
6641:0
6637:y
6633:,
6628:0
6624:x
6620:(
6600:)
6595:n
6591:y
6587:,
6582:n
6578:x
6574:(
6552:.
6549:)
6544:i
6540:y
6531:1
6528:+
6525:i
6521:x
6512:1
6509:+
6506:i
6502:y
6493:i
6489:x
6485:(
6480:1
6474:n
6469:0
6466:=
6463:i
6453:2
6450:1
6445:=
6442:A
6416:A
6394:,
6391:)
6386:i
6382:y
6373:1
6370:+
6367:i
6363:x
6354:1
6351:+
6348:i
6344:y
6335:i
6331:x
6327:(
6324:)
6319:1
6316:+
6313:i
6309:y
6305:+
6300:i
6296:y
6292:(
6287:1
6281:n
6276:0
6273:=
6270:i
6259:A
6256:6
6252:1
6247:=
6241:y
6236:C
6211:,
6208:)
6203:i
6199:y
6190:1
6187:+
6184:i
6180:x
6171:1
6168:+
6165:i
6161:y
6152:i
6148:x
6144:(
6141:)
6136:1
6133:+
6130:i
6126:x
6122:+
6117:i
6113:x
6109:(
6104:1
6098:n
6093:0
6090:=
6087:i
6076:A
6073:6
6069:1
6064:=
6058:x
6053:C
6030:,
6027:)
6022:y
6018:C
6014:,
6009:x
6005:C
6001:(
5981:,
5978:)
5973:1
5967:n
5963:y
5959:,
5954:1
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5944:x
5940:(
5920:,
5913:,
5910:)
5905:1
5901:y
5897:,
5892:1
5888:x
5884:(
5864:,
5861:)
5856:0
5852:y
5848:,
5843:0
5839:x
5835:(
5815:n
5784:.
5781:G
5778:P
5775:3
5772:+
5769:)
5766:F
5763:P
5760:+
5757:E
5754:P
5751:+
5748:D
5745:P
5742:(
5739:2
5733:C
5730:P
5727:+
5724:B
5721:P
5718:+
5715:A
5712:P
5691:,
5688:C
5685:B
5682:A
5662:P
5642:,
5639:B
5636:A
5633:,
5630:A
5627:C
5624:,
5621:C
5618:B
5598:F
5595:,
5592:E
5589:,
5586:D
5566:G
5546:C
5543:B
5540:A
5515:.
5512:)
5507:2
5503:C
5499:G
5496:+
5491:2
5487:B
5483:G
5480:+
5475:2
5471:A
5467:G
5464:(
5461:3
5458:=
5453:2
5449:A
5445:C
5442:+
5437:2
5433:C
5429:B
5426:+
5421:2
5417:B
5413:A
5389:.
5384:2
5380:G
5376:P
5373:3
5370:+
5365:2
5361:C
5357:G
5354:+
5349:2
5345:B
5341:G
5338:+
5333:2
5329:A
5325:G
5322:=
5317:2
5313:C
5309:P
5306:+
5301:2
5297:B
5293:P
5290:+
5285:2
5281:A
5277:P
5266::
5254:G
5233:P
5213:G
5193:P
5173:.
5170:G
5150:C
5147:,
5144:B
5141:,
5138:A
5118:P
5092:9
5089:5
5048:.
5045:)
5042:C
5039:B
5036:A
5025:(
5019:3
5016:1
5010:=
5007:)
5004:G
5001:C
4998:B
4987:(
4984:=
4981:)
4978:G
4975:C
4972:A
4961:(
4958:=
4955:)
4952:G
4949:B
4946:A
4935:(
4913:,
4910:C
4907:B
4904:A
4884:G
4858:.
4849:O
4846:I
4827:C
4824:I
4814:,
4805:C
4802:H
4783:H
4780:I
4770:,
4761:C
4758:H
4739:C
4736:I
4726:,
4717:N
4714:C
4708:2
4705:=
4692:O
4689:C
4679:,
4670:N
4667:C
4661:4
4658:=
4645:H
4642:C
4615:,
4612:N
4589:I
4564:.
4555:O
4552:C
4546:2
4543:=
4534:H
4531:C
4506:,
4503:O
4480:H
4447:2
4444:3
4395:.
4392:M
4380:L
4371:+
4368:N
4359::
4356:L
4344:N
4335:+
4332:M
4323::
4320:N
4308:M
4299:+
4296:L
4287:=
4277:M
4265:L
4256:+
4253:N
4244::
4241:L
4229:N
4220:+
4217:M
4208::
4205:N
4193:M
4184:+
4181:L
4172:=
4162:N
4153::
4150:M
4141::
4138:L
4129:=
4126:b
4123:a
4120::
4117:a
4114:c
4111::
4108:c
4105:b
4102:=
4097:c
4094:1
4089::
4084:b
4081:1
4076::
4071:a
4068:1
4063:=
4056:C
4041::
4029:N
4026:,
4023:M
4020:,
4017:L
3996:c
3993:,
3990:b
3987:,
3984:a
3950:3
3947:1
3941::
3935:3
3932:1
3926::
3920:3
3917:1
3892:.
3887:)
3882:)
3877:N
3873:y
3869:+
3864:M
3860:y
3856:+
3851:L
3847:y
3843:(
3837:3
3834:1
3828:,
3825:)
3820:N
3816:x
3812:+
3807:M
3803:x
3799:+
3794:L
3790:x
3786:(
3780:3
3777:1
3769:(
3764:=
3761:)
3758:N
3755:+
3752:M
3749:+
3746:L
3743:(
3737:3
3734:1
3728:=
3725:C
3699:G
3679:C
3659:,
3656:)
3651:N
3647:y
3643:,
3638:N
3634:x
3630:(
3627:=
3624:N
3604:,
3601:)
3596:M
3592:y
3588:,
3583:M
3579:x
3575:(
3572:=
3569:M
3550:,
3547:)
3542:L
3538:y
3534:,
3529:L
3525:x
3521:(
3518:=
3515:L
3484:3
3481:1
3458:,
3455:1
3452::
3449:2
3381:O
3361:.
3358:O
3338:,
3335:D
3332:C
3312:B
3309:A
3287:.
3284:D
3281:C
3258:.
3255:B
3252:A
3192:x
3189:d
3184:)
3179:)
3176:x
3173:(
3170:g
3164:)
3161:x
3158:(
3155:f
3150:(
3143:b
3138:a
3113:A
3087:,
3084:x
3081:d
3075:)
3070:)
3067:x
3064:(
3061:g
3055:)
3052:x
3049:(
3046:f
3041:(
3034:)
3029:)
3026:x
3023:(
3020:g
3017:+
3014:)
3011:x
3008:(
3005:f
3000:(
2992:2
2989:1
2981:b
2976:a
2966:A
2963:1
2958:=
2945:y
2935:,
2932:x
2929:d
2923:)
2918:)
2915:x
2912:(
2909:g
2903:)
2900:x
2897:(
2894:f
2889:(
2884:x
2879:b
2874:a
2864:A
2861:1
2856:=
2843:x
2814:b
2808:x
2802:a
2783:,
2780:]
2777:b
2774:,
2771:a
2768:[
2748:)
2745:x
2742:(
2739:g
2733:)
2730:x
2727:(
2724:f
2704:g
2684:f
2661:)
2652:y
2645:,
2636:x
2630:(
2603:.
2600:y
2577:X
2557:)
2554:y
2551:(
2545:x
2540:S
2519:,
2516:x
2493:X
2473:)
2470:x
2467:(
2461:y
2456:S
2436:,
2433:X
2413:A
2391:,
2386:A
2382:y
2379:d
2373:)
2370:y
2367:(
2361:x
2356:S
2352:y
2343:=
2337:y
2332:C
2327:,
2322:A
2318:x
2315:d
2309:)
2306:x
2303:(
2297:y
2292:S
2288:x
2279:=
2273:x
2268:C
2243:.
2240:X
2220:.
2217:z
2214:=
2209:k
2205:x
2184:X
2164:)
2161:z
2158:(
2153:k
2149:S
2128:,
2125:C
2105:k
2083:k
2079:C
2056:,
2050:x
2047:d
2041:)
2038:x
2035:(
2032:g
2024:z
2021:d
2015:)
2012:z
2009:(
2004:k
2000:S
1996:z
1987:=
1982:k
1978:C
1952:X
1932:.
1929:X
1905:0
1902:=
1899:)
1896:x
1893:(
1890:g
1870:X
1864:x
1844:1
1841:=
1838:)
1835:x
1832:(
1829:g
1823::
1817:n
1812:R
1790:X
1766:g
1746:,
1741:n
1736:R
1705:x
1702:d
1696:)
1693:x
1690:(
1687:g
1679:x
1676:d
1670:)
1667:x
1664:(
1661:g
1658:x
1649:=
1646:C
1622:n
1617:R
1595:X
1563:d
1543:,
1540:d
1520:,
1515:d
1510:R
1488:,
1483:i
1479:X
1456:i
1452:A
1428:.
1409:2
1393:2
1383:2
1380:1
1375:+
1370:2
1358:2
1344:3
1336:2
1326:2
1323:1
1312:+
1307:2
1296:5
1290:=
1287:x
1215:0
1195:,
1192:X
1172:p
1152:1
1132:p
1110:i
1106:A
1083:i
1079:A
1058:.
1053:i
1049:A
1028:,
1025:X
1003:.
995:i
991:A
985:i
973:i
969:A
963:y
956:i
952:C
944:i
933:=
928:y
924:C
919:,
911:i
907:A
901:i
889:i
885:A
879:x
872:i
868:C
860:i
849:=
844:x
840:C
815:i
811:A
788:i
784:C
763:,
758:n
754:X
750:,
744:,
739:2
735:X
731:,
726:1
722:X
701:X
673:.
668:k
662:k
657:x
652:+
646:+
641:2
636:x
631:+
626:1
621:x
613:=
609:C
586:n
581:R
557:k
552:x
547:,
541:,
536:2
531:x
526:,
521:1
516:x
494:k
152:-
140:n
83:)
77:(
72:)
68:(
54:.
20:)
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