2651:
1799:
1493:
1133:
1300:
2722:
to the image in the image (sensor) plane. The three points, object point, image point and optical centre, are always collinear. Another way to say this is that the line segments joining the object points with their image points are all concurrent at the optical centre.
2598:
1564:
2427:
1325:
554:
895:
102:, where lines are represented in the standard model by great circles of a sphere, sets of collinear points lie on the same great circle. Such points do not lie on a "straight line" in the Euclidean sense, and are not thought of as being
1147:
2436:. In practice, we rarely face perfect multicollinearity in a data set. More commonly, the issue of multicollinearity arises when there is a "strong linear relationship" among two or more independent variables, meaning that
97:
for the geometry offers an interpretation of how the points, lines and other object types relate to one another and a notion such as collinearity must be interpreted within the context of that model. For instance, in
766:
states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line, then the six vertices of the hexagon lie on a conic, which may be degenerate as in
762:, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon. The converse is also true: the
2209:
900:
2442:
1983:. Given a set of collinear points, by plane duality we obtain a set of lines all of which meet at a common point. The property that this set of lines has (meeting at a common point) is called
789:
in a plane, none of which is completely inside one of the others, the three intersection points of the three pairs of lines, each externally tangent to two of the circles, are collinear.
1794:{\displaystyle \det {\begin{bmatrix}0&d(AB)^{2}&d(AC)^{2}&1\\d(AB)^{2}&0&d(BC)^{2}&1\\d(AC)^{2}&d(BC)^{2}&0&1\\1&1&1&0\end{bmatrix}}=0.}
1517:
with those three points as vertices, this is equivalent to the statement that the three points are collinear if and only if the triangle with those points as vertices has zero area.
2284:
1488:{\displaystyle {\begin{bmatrix}1&x_{1}&x_{2}&\dots &x_{n}\\1&y_{1}&y_{2}&\dots &y_{n}\\1&z_{1}&z_{2}&\dots &z_{n}\end{bmatrix}}}
393:
336:
2628:
404:
2130:
2103:
1128:{\displaystyle {\begin{aligned}X&=(x_{1},\ x_{2},\ \dots ,\ x_{n}),\\Y&=(y_{1},\ y_{2},\ \dots ,\ y_{n}),\\Z&=(z_{1},\ z_{2},\ \dots ,\ z_{n}),\end{aligned}}}
1295:{\displaystyle {\begin{bmatrix}x_{1}&x_{2}&\dots &x_{n}\\y_{1}&y_{2}&\dots &y_{n}\\z_{1}&z_{2}&\dots &z_{n}\end{bmatrix}}}
3062:
2706:) plane (in two dimensions) to object coordinates (in three dimensions). In the photography setting, the equations are derived by considering the
885:-dimensional space, a set of three or more distinct points are collinear if and only if, the matrix of the coordinates of these vectors is of
123:, viewed as geometric maps, map lines to lines; that is, they map collinear point sets to collinear point sets and so, are collineations. In
85:
this relation is intuitively visualized by points lying in a row on a "straight line". However, in most geometries (including
Euclidean) a
203:
The midpoint of any side, the point that is equidistant from it along the triangle's boundary in either direction (so these two points
712:
701:
673:
566:
833:
of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.
2060:
if there is an exact linear relationship between the two, so the correlation between them is equal to 1 or −1. That is,
2914:
2715:
2142:
2637:
expands on this traditional view, and refers to collinearity between explanatory and criteria (i.e., explained) variables.
2593:{\displaystyle X_{ki}=\lambda _{0}+\lambda _{1}X_{1i}+\lambda _{2}X_{2i}+\dots +\lambda _{k-1}X_{(k-1),i}+\varepsilon _{i}}
705:
1979:
the notion of interchanging the roles of "points" and "lines" while preserving the relationship between them is called
763:
69:). In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".
3070:
3041:
3020:
860:
844:
805:
1844:
Equivalently, a set of at least three distinct points are collinear if and only if, for every three of those points
686:
1526:
182:
1980:
1534:
204:
2015:
1837:; so checking if this determinant equals zero is equivalent to checking whether the triangle with vertices
253:
of a triangle, the nearest points on each of the three extended sides of the triangle are collinear in the
243:
2422:{\displaystyle X_{ki}=\lambda _{0}+\lambda _{1}X_{1i}+\lambda _{2}X_{2i}+\dots +\lambda _{k-1}X_{(k-1),i}}
1931:—that is, they share a common factor other than 1—if and only if for a rectangle plotted on a
3096:
2757:
768:
2950:"Lateral collinearity and misleading results in variance-based SEM: An illustration and recommendations"
2747:
742:(also known as the Hexagrammum Mysticum Theorem) states that if an arbitrary six points are chosen on a
808:
are collinear, and the center and the two vertices with the greatest radius of curvature are collinear.
345:
288:
2606:
549:{\displaystyle P_{1}A_{2}\cdot P_{2}A_{3}\cdot P_{3}A_{1}=P_{1}A_{3}\cdot P_{2}A_{1}\cdot P_{3}A_{2}.}
17:
2752:
2707:
682:
640:
2108:
2081:
2824:
2663:
589:
2695:
395:
respectively are collinear if and only if the following products of segment lengths are equal:
2902:
2687:
1499:
1306:
886:
3054:
2949:
3080:
2844:
2742:
2053:
1317:
1139:
719:
693:
282:
8:
2275:
1882:
878:
739:
577:
272:
261:
197:
167:
124:
1529:, meaning all the points are collinear, if and only if, for every three of those points
588:. The midpoints on the three sides of these points of intersection are collinear in the
3055:
2972:
1807:
782:
212:
99:
94:
82:
2909:, New Mathematical Library, vol. 37, Cambridge University Press, pp. 35–39,
2872:
3066:
3037:
3016:
2910:
2828:
2797:
2659:
2039:
1514:
729:
In a tangential trapezoid, the midpoints of the legs are collinear with the incenter.
171:
39:
3029:
2976:
2964:
2842:
Scott, J. A. "Some examples of the use of areal coordinates in triangle geometry",
2732:
2000:
1988:
1976:
1502:
2 or less, the points are collinear. In particular, for three points in the plane (
797:
90:
54:
2780:, pg. 26), but is often only defined within the discussion of a specific geometry
1810:, equal to −16 times the square of the area of a triangle with side lengths
3076:
3050:
2930:
2898:
2675:
2023:
1928:
224:
86:
58:
2678:
are parallel and aligned, that is they are located along a common line or axis.
2650:
2256:
plane, these points are collinear in the sense defined earlier in this article.
2719:
2691:
2671:
1932:
826:
801:
232:
228:
216:
208:
1509:), the above matrix is square and the points are collinear if and only if its
3090:
2667:
743:
603:
585:
570:
339:
120:
1520:
1513:
is zero; since that 3 Ă— 3 determinant is plus or minus twice the
848:
715:, the vertex centroid, and the intersection of the diagonals are collinear.
697:
250:
163:
155:
111:
35:
562:
The incenter, the centroid, and the
Spieker circle's center are collinear.
3008:
2968:
2737:
2711:
2699:
1510:
636:
581:
254:
242:
Any vertex, the tangency of the opposite side with the incircle, and the
193:
151:
109:
A mapping of a geometry to itself which sends lines to lines is called a
2045:
856:
175:
129:
116:
872:
812:
755:
236:
751:
723:
674:
Quadrilateral#Remarkable points and lines in a convex quadrilateral
618:
276:
268:
220:
189:
159:
46:
2674:
mounted in such a manner that the corresponding elements of each
793:
759:
747:
31:
2989:
It's more mathematically natural to refer to these equations as
93:, so such visualizations will not necessarily be appropriate. A
2907:
Episodes in
Nineteenth and Twentieth Century Euclidean Geometry
2703:
815:, the center, the two foci, and the two vertices are collinear.
786:
2993:, but photogrammetry literature does not use that terminology.
2857:
Dušan Djukić, Vladimir
Janković, Ivan Matić, Nikola Petrović,
1991:. Thus, concurrency is the plane dual notion to collinearity.
275:, and the point of contact of the corresponding side with the
146:
In any triangle the following sets of points are collinear:
77:
In any geometry, the set of points on a line are said to be
830:
843:
The centroid of a tetrahedron is the midpoint between its
1521:
Collinearity of points whose pairwise distances are given
30:"Colinear" redirects here. For the use in genetics, see
2654:
An antenna mast with four collinear directional arrays.
2204:{\displaystyle X_{2i}=\lambda _{0}+\lambda _{1}X_{1i}.}
2033:
1576:
1334:
1156:
635:
are collinear and the line through them is called the
188:
Any vertex, the tangency of the opposite side with an
2609:
2445:
2287:
2145:
2111:
2084:
1567:
1328:
1150:
898:
758:) and joined by line segments in any order to form a
407:
348:
291:
2873:"On Two Remarkable Lines Related to a Quadrilateral"
584:
of a triangle each intersect each of the triangle's
2278:model are perfectly linearly related, according to
726:
with the two bases are collinear with the incenter.
61:. A set of points with this property is said to be
2957:Journal of the Association for Information Systems
2622:
2592:
2421:
2203:
2124:
2097:
2078:are perfectly collinear if there exist parameters
1793:
1525:A set of at least three distinct points is called
1487:
1294:
1127:
873:Collinearity of points whose coordinates are given
548:
387:
330:
3006:
2932:Three Centroids created by a Cyclic Quadrilateral
2785:
2006:, where two points determine at most one line, a
704:(the intersection of the two bimedians), and the
264:intersect the opposite sides at collinear points.
136:
3088:
3063:Ergebnisse der Mathematik und ihrer Grenzgebiete
1955:, at least one interior point is collinear with
1568:
646:In a convex quadrilateral, the quasiorthocenter
174:are collinear, all falling on a line called the
1841:has zero area (so the vertices are collinear).
115:; it preserves the collinearity property. The
889:1 or less. For example, given three points
27:Property of points all lying on a single line
2214:This means that if the various observations
2052:refers to a linear relationship between two
855:of the tetrahedron that is analogous to the
1970:
643:, then its incenter also lies on this line.
57:is the property of their lying on a single
2897:
3049:
2891:
2870:
2807:
2805:
2777:
687:Tangential quadrilateral#Collinear points
2947:
2864:
2649:
2026:if and only if they determine a line in
3028:
3007:Brannan, David A.; Esplen, Matthew F.;
2928:
2781:
2640:
133:and are just one type of collineation.
34:. For the use in coalgebra theory, see
14:
3089:
2802:
2664:collinear (or co-linear) antenna array
2833:, 2nd ed. Barnes & Noble, 1952 .
2798:Colinear (Merriam-Webster dictionary)
1994:
1309:1 or less, the points are collinear.
260:The lines connecting the feet of the
38:. For colinearity in statistics, see
2776:The concept applies in any geometry
2690:are a set of two equations, used in
2034:Usage in statistics and econometrics
279:relative to that side are collinear.
3036:, New York: John Wiley & Sons,
72:
24:
3065:, vol. 44, Berlin: Springer,
1312:Equivalently, for every subset of
609:whose opposite sides intersect at
342:) of a triangle opposite vertices
25:
3108:
2818:
2815:, Dover Publ., 2007 (orig. 1929).
2748:Incidence (geometry)#Collinearity
2645:
2018:whose vertices are the points of
1859:greater than or equal to each of
1533:, the following determinant of a
861:tetrahedron's twelve-point sphere
859:of a triangle. The center of the
775:
658:are collinear in this order, and
596:
388:{\displaystyle A_{1},A_{2},A_{3}}
331:{\displaystyle P_{1},P_{2},P_{3}}
257:of the point on the circumcircle.
211:are collinear in a line called a
196:are collinear in a line called a
181:The de Longchamps point also has
127:these linear mappings are called
117:linear maps (or linear functions)
91:primitive (undefined) object type
2786:Brannan, Esplen & Gray (1998
2623:{\displaystyle \varepsilon _{i}}
2132:such that, for all observations
1914:
2983:
2941:
2263:refers to a situation in which
1987:, and the lines are said to be
837:
711:In a cyclic quadrilateral, the
3015:, Cambridge University Press,
2948:Kock, N.; Lynn, G. S. (2012).
2922:
2851:
2836:
2791:
2770:
2681:
2566:
2554:
2408:
2396:
1736:
1726:
1712:
1702:
1681:
1671:
1652:
1642:
1621:
1611:
1597:
1587:
1115:
1061:
1041:
987:
967:
913:
764:Braikenridge–Maclaurin theorem
137:Examples in Euclidean geometry
13:
1:
3000:
2929:Bradley, Christopher (2011),
1548:meaning the distance between
2125:{\displaystyle \lambda _{1}}
2098:{\displaystyle \lambda _{0}}
863:also lies on the Euler line.
654:, and the quasicircumcenter
639:. If the quadrilateral is a
573:of a triangle are collinear.
209:center of the Spieker circle
141:
7:
2903:"4.2 Cyclic quadrilaterals"
2848:83, November 1999, 472–477.
2813:Advanced Euclidean Geometry
2726:
2274:explanatory variables in a
733:
10:
3113:
2871:Myakishev, Alexei (2006),
2037:
867:
851:. These points define the
681:Other collinearities of a
29:
2022:, where two vertices are
1535:Cayley–Menger determinant
285:states that three points
3034:Introduction to Geometry
2861:, Springer, 2006, p. 15.
2825:Altshiller Court, Nathan
2763:
2758:Pappus's hexagon theorem
2753:No-three-in-line problem
1971:Concurrency (plane dual)
1806:This determinant is, by
819:
769:Pappus's hexagon theorem
722:, the tangencies of the
683:tangential quadrilateral
641:tangential quadrilateral
170:, and the center of the
2696:computer stereo vision
2688:collinearity equations
2655:
2624:
2603:where the variance of
2594:
2423:
2205:
2126:
2099:
1795:
1489:
1296:
1129:
796:, the center, the two
650:, the "area centroid"
565:The circumcenter, the
550:
389:
332:
249:From any point on the
215:of the triangle. (The
65:(sometimes spelled as
2991:concurrency equations
2653:
2630:is relatively small.
2625:
2595:
2432:for all observations
2424:
2206:
2127:
2100:
2054:explanatory variables
1911:holds with equality.
1796:
1490:
1297:
1130:
551:
390:
333:
271:, the midpoint of an
2969:10.17705/1jais.00302
2845:Mathematical Gazette
2743:Direction (geometry)
2641:Usage in other areas
2635:lateral collinearity
2607:
2443:
2285:
2143:
2109:
2082:
2056:. Two variables are
1565:
1326:
1148:
896:
720:tangential trapezoid
694:cyclic quadrilateral
580:intersecting at the
405:
346:
338:on the sides (some
289:
205:bisect the perimeter
183:other collinearities
2880:Forum Geometricorum
2811:Johnson, Roger A.,
2276:multiple regression
2238:are plotted in the
2058:perfectly collinear
1883:triangle inequality
879:coordinate geometry
806:radius of curvature
578:perpendicular lines
168:de Longchamps point
125:projective geometry
3097:Incidence geometry
2859:The IMO Compendium
2710:of a point of the
2708:central projection
2660:telecommunications
2656:
2620:
2590:
2419:
2201:
2122:
2095:
2008:collinearity graph
1995:Collinearity graph
1791:
1779:
1515:area of a triangle
1485:
1479:
1292:
1286:
1125:
1123:
804:with the smallest
546:
385:
328:
100:spherical geometry
83:Euclidean geometry
3057:Finite geometries
3030:Coxeter, H. S. M.
2916:978-0-88385-639-0
2261:multicollinearity
2040:Multicollinearity
1935:with vertices at
1104:
1095:
1079:
1030:
1021:
1005:
956:
947:
931:
283:Menelaus' theorem
239:of the triangle.)
172:nine-point circle
40:multicollinearity
16:(Redirected from
3104:
3083:
3060:
3051:Dembowski, Peter
3046:
3025:
2994:
2987:
2981:
2980:
2954:
2945:
2939:
2938:
2937:
2926:
2920:
2919:
2899:Honsberger, Ross
2895:
2889:
2887:
2877:
2868:
2862:
2855:
2849:
2840:
2834:
2830:College Geometry
2822:
2816:
2809:
2800:
2795:
2789:
2774:
2733:Concyclic points
2629:
2627:
2626:
2621:
2619:
2618:
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2588:
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2547:
2523:
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2510:
2509:
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2435:
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2273:
2255:
2237:
2210:
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2202:
2197:
2196:
2184:
2183:
2171:
2170:
2158:
2157:
2135:
2131:
2129:
2128:
2123:
2121:
2120:
2104:
2102:
2101:
2096:
2094:
2093:
2077:
2068:
2029:
2021:
2013:
2005:
2001:partial geometry
1989:concurrent lines
1977:plane geometries
1966:
1958:
1954:
1926:
1922:
1910:
1880:
1869:
1858:
1847:
1840:
1836:
1800:
1798:
1797:
1792:
1784:
1783:
1744:
1743:
1720:
1719:
1689:
1688:
1660:
1659:
1629:
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1605:
1604:
1555:
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1547:
1532:
1508:
1494:
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1484:
1483:
1476:
1475:
1459:
1458:
1447:
1446:
1428:
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1411:
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1399:
1398:
1380:
1379:
1363:
1362:
1351:
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1315:
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1299:
1298:
1293:
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1254:
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1180:
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1168:
1167:
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1114:
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1040:
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1003:
999:
998:
966:
965:
954:
945:
941:
940:
929:
925:
924:
884:
785:, for any three
740:Pascal's theorem
671:
670:
664:
657:
653:
649:
634:
633:
629:
625:
616:
612:
608:
567:Brocard midpoint
555:
553:
552:
547:
542:
541:
532:
531:
519:
518:
509:
508:
496:
495:
486:
485:
473:
472:
463:
462:
450:
449:
440:
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426:
417:
416:
394:
392:
391:
386:
384:
383:
371:
370:
358:
357:
337:
335:
334:
329:
327:
326:
314:
313:
301:
300:
200:of the triangle.
73:Points on a line
21:
3112:
3111:
3107:
3106:
3105:
3103:
3102:
3101:
3087:
3086:
3073:
3044:
3023:
3009:Gray, Jeremy J.
3003:
2998:
2997:
2988:
2984:
2952:
2946:
2942:
2935:
2927:
2923:
2917:
2896:
2892:
2875:
2869:
2865:
2856:
2852:
2841:
2837:
2823:
2819:
2810:
2803:
2796:
2792:
2778:Dembowski (1968
2775:
2771:
2766:
2729:
2684:
2672:dipole antennas
2648:
2643:
2633:The concept of
2614:
2610:
2608:
2605:
2604:
2584:
2580:
2553:
2549:
2537:
2533:
2515:
2511:
2505:
2501:
2489:
2485:
2479:
2475:
2466:
2462:
2450:
2446:
2444:
2441:
2440:
2433:
2395:
2391:
2379:
2375:
2357:
2353:
2347:
2343:
2331:
2327:
2321:
2317:
2308:
2304:
2292:
2288:
2286:
2283:
2282:
2264:
2253:
2246:
2239:
2235:
2225:
2215:
2189:
2185:
2179:
2175:
2166:
2162:
2150:
2146:
2144:
2141:
2140:
2133:
2116:
2112:
2110:
2107:
2106:
2089:
2085:
2083:
2080:
2079:
2076:
2070:
2067:
2061:
2042:
2036:
2027:
2019:
2011:
2003:
1997:
1973:
1960:
1956:
1936:
1924:
1920:
1917:
1885:
1871:
1860:
1849:
1845:
1838:
1811:
1808:Heron's formula
1778:
1777:
1772:
1767:
1762:
1756:
1755:
1750:
1745:
1739:
1735:
1721:
1715:
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1666:
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1655:
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1635:
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1624:
1620:
1606:
1600:
1596:
1582:
1572:
1571:
1566:
1563:
1562:
1553:
1549:
1538:
1530:
1523:
1503:
1478:
1477:
1471:
1467:
1465:
1460:
1454:
1450:
1448:
1442:
1438:
1436:
1430:
1429:
1423:
1419:
1417:
1412:
1406:
1402:
1400:
1394:
1390:
1388:
1382:
1381:
1375:
1371:
1369:
1364:
1358:
1354:
1352:
1346:
1342:
1340:
1330:
1329:
1327:
1324:
1323:
1313:
1285:
1284:
1278:
1274:
1272:
1267:
1261:
1257:
1255:
1249:
1245:
1242:
1241:
1235:
1231:
1229:
1224:
1218:
1214:
1212:
1206:
1202:
1199:
1198:
1192:
1188:
1186:
1181:
1175:
1171:
1169:
1163:
1159:
1152:
1151:
1149:
1146:
1145:
1122:
1121:
1109:
1105:
1084:
1080:
1068:
1064:
1054:
1048:
1047:
1035:
1031:
1010:
1006:
994:
990:
980:
974:
973:
961:
957:
936:
932:
920:
916:
906:
899:
897:
894:
893:
882:
875:
870:
840:
822:
783:Monge's theorem
778:
736:
702:vertex centroid
666:
660:
659:
655:
651:
647:
631:
627:
623:
622:
614:
610:
606:
599:
590:Droz–Farny line
537:
533:
527:
523:
514:
510:
504:
500:
491:
487:
481:
477:
468:
464:
458:
454:
445:
441:
435:
431:
422:
418:
412:
408:
406:
403:
402:
379:
375:
366:
362:
353:
349:
347:
344:
343:
322:
318:
309:
305:
296:
292:
290:
287:
286:
225:medial triangle
144:
139:
89:is typically a
75:
43:
28:
23:
22:
15:
12:
11:
5:
3110:
3100:
3099:
3085:
3084:
3071:
3047:
3042:
3026:
3021:
3002:
2999:
2996:
2995:
2982:
2963:(7): 546–580.
2940:
2921:
2915:
2890:
2863:
2850:
2835:
2817:
2801:
2790:
2768:
2767:
2765:
2762:
2761:
2760:
2755:
2750:
2745:
2740:
2735:
2728:
2725:
2716:optical centre
2692:photogrammetry
2683:
2680:
2647:
2646:Antenna arrays
2644:
2642:
2639:
2617:
2613:
2601:
2600:
2587:
2583:
2579:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2552:
2546:
2543:
2540:
2536:
2532:
2529:
2526:
2521:
2518:
2514:
2508:
2504:
2500:
2495:
2492:
2488:
2482:
2478:
2474:
2469:
2465:
2461:
2456:
2453:
2449:
2430:
2429:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2394:
2388:
2385:
2382:
2378:
2374:
2371:
2368:
2363:
2360:
2356:
2350:
2346:
2342:
2337:
2334:
2330:
2324:
2320:
2316:
2311:
2307:
2303:
2298:
2295:
2291:
2251:
2244:
2230:
2220:
2212:
2211:
2200:
2195:
2192:
2188:
2182:
2178:
2174:
2169:
2165:
2161:
2156:
2153:
2149:
2119:
2115:
2092:
2088:
2074:
2065:
2038:Main article:
2035:
2032:
1996:
1993:
1972:
1969:
1933:square lattice
1916:
1913:
1804:
1803:
1802:
1801:
1790:
1787:
1782:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1758:
1757:
1754:
1751:
1749:
1746:
1742:
1738:
1734:
1731:
1728:
1725:
1722:
1718:
1714:
1710:
1707:
1704:
1701:
1698:
1697:
1694:
1691:
1687:
1683:
1679:
1676:
1673:
1670:
1667:
1665:
1662:
1658:
1654:
1650:
1647:
1644:
1641:
1638:
1637:
1634:
1631:
1627:
1623:
1619:
1616:
1613:
1610:
1607:
1603:
1599:
1595:
1592:
1589:
1586:
1583:
1581:
1578:
1577:
1575:
1570:
1537:is zero (with
1522:
1519:
1496:
1495:
1482:
1474:
1470:
1466:
1464:
1461:
1457:
1453:
1449:
1445:
1441:
1437:
1435:
1432:
1431:
1426:
1422:
1418:
1416:
1413:
1409:
1405:
1401:
1397:
1393:
1389:
1387:
1384:
1383:
1378:
1374:
1370:
1368:
1365:
1361:
1357:
1353:
1349:
1345:
1341:
1339:
1336:
1335:
1333:
1303:
1302:
1289:
1281:
1277:
1273:
1271:
1268:
1264:
1260:
1256:
1252:
1248:
1244:
1243:
1238:
1234:
1230:
1228:
1225:
1221:
1217:
1213:
1209:
1205:
1201:
1200:
1195:
1191:
1187:
1185:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1157:
1155:
1136:
1135:
1120:
1117:
1112:
1108:
1101:
1098:
1092:
1087:
1083:
1076:
1071:
1067:
1063:
1060:
1057:
1055:
1053:
1050:
1049:
1046:
1043:
1038:
1034:
1027:
1024:
1018:
1013:
1009:
1002:
997:
993:
989:
986:
983:
981:
979:
976:
975:
972:
969:
964:
960:
953:
950:
944:
939:
935:
928:
923:
919:
915:
912:
909:
907:
905:
902:
901:
874:
871:
869:
866:
865:
864:
839:
836:
835:
834:
827:center of mass
821:
818:
817:
816:
809:
800:, and the two
790:
777:
776:Conic sections
774:
773:
772:
735:
732:
731:
730:
727:
716:
709:
708:are collinear.
690:
678:
677:
644:
598:
597:Quadrilaterals
595:
594:
593:
586:extended sides
574:
563:
559:
558:
557:
556:
545:
540:
536:
530:
526:
522:
517:
513:
507:
503:
499:
494:
490:
484:
480:
476:
471:
467:
461:
457:
453:
448:
444:
438:
434:
430:
425:
421:
415:
411:
397:
396:
382:
378:
374:
369:
365:
361:
356:
352:
325:
321:
317:
312:
308:
304:
299:
295:
280:
265:
258:
247:
246:are collinear.
244:Gergonne point
240:
233:center of mass
217:Spieker circle
201:
186:
179:
143:
140:
138:
135:
74:
71:
26:
9:
6:
4:
3:
2:
3109:
3098:
3095:
3094:
3092:
3082:
3078:
3074:
3072:3-540-61786-8
3068:
3064:
3059:
3058:
3052:
3048:
3045:
3043:0-471-50458-0
3039:
3035:
3031:
3027:
3024:
3022:0-521-59787-0
3018:
3014:
3010:
3005:
3004:
2992:
2986:
2978:
2974:
2970:
2966:
2962:
2958:
2951:
2944:
2934:
2933:
2925:
2918:
2912:
2908:
2904:
2900:
2894:
2885:
2881:
2874:
2867:
2860:
2854:
2847:
2846:
2839:
2832:
2831:
2826:
2821:
2814:
2808:
2806:
2799:
2794:
2787:
2783:
2782:Coxeter (1969
2779:
2773:
2769:
2759:
2756:
2754:
2751:
2749:
2746:
2744:
2741:
2739:
2736:
2734:
2731:
2730:
2724:
2721:
2717:
2713:
2709:
2705:
2702:in an image (
2701:
2697:
2693:
2689:
2679:
2677:
2673:
2669:
2665:
2661:
2652:
2638:
2636:
2631:
2615:
2611:
2585:
2581:
2577:
2572:
2569:
2563:
2560:
2557:
2550:
2544:
2541:
2538:
2534:
2530:
2527:
2524:
2519:
2516:
2512:
2506:
2502:
2498:
2493:
2490:
2486:
2480:
2476:
2472:
2467:
2463:
2459:
2454:
2451:
2447:
2439:
2438:
2437:
2414:
2411:
2405:
2402:
2399:
2392:
2386:
2383:
2380:
2376:
2372:
2369:
2366:
2361:
2358:
2354:
2348:
2344:
2340:
2335:
2332:
2328:
2322:
2318:
2314:
2309:
2305:
2301:
2296:
2293:
2289:
2281:
2280:
2279:
2277:
2271:
2267:
2262:
2257:
2250:
2243:
2234:
2229:
2224:
2219:
2198:
2193:
2190:
2186:
2180:
2176:
2172:
2167:
2163:
2159:
2154:
2151:
2147:
2139:
2138:
2137:
2117:
2113:
2090:
2086:
2073:
2064:
2059:
2055:
2051:
2047:
2041:
2031:
2025:
2017:
2009:
2002:
1992:
1990:
1986:
1982:
1981:plane duality
1978:
1968:
1964:
1952:
1948:
1944:
1940:
1934:
1930:
1915:Number theory
1912:
1908:
1904:
1900:
1896:
1892:
1888:
1884:
1878:
1874:
1867:
1863:
1856:
1852:
1842:
1834:
1830:
1826:
1822:
1818:
1814:
1809:
1788:
1785:
1780:
1774:
1769:
1764:
1759:
1752:
1747:
1740:
1732:
1729:
1723:
1716:
1708:
1705:
1699:
1692:
1685:
1677:
1674:
1668:
1663:
1656:
1648:
1645:
1639:
1632:
1625:
1617:
1614:
1608:
1601:
1593:
1590:
1584:
1579:
1573:
1561:
1560:
1559:
1558:
1557:
1545:
1541:
1536:
1528:
1518:
1516:
1512:
1506:
1501:
1480:
1472:
1468:
1462:
1455:
1451:
1443:
1439:
1433:
1424:
1420:
1414:
1407:
1403:
1395:
1391:
1385:
1376:
1372:
1366:
1359:
1355:
1347:
1343:
1337:
1331:
1322:
1321:
1320:
1319:
1310:
1308:
1287:
1279:
1275:
1269:
1262:
1258:
1250:
1246:
1236:
1232:
1226:
1219:
1215:
1207:
1203:
1193:
1189:
1183:
1176:
1172:
1164:
1160:
1153:
1144:
1143:
1142:
1141:
1118:
1110:
1106:
1099:
1096:
1090:
1085:
1081:
1074:
1069:
1065:
1058:
1056:
1051:
1044:
1036:
1032:
1025:
1022:
1016:
1011:
1007:
1000:
995:
991:
984:
982:
977:
970:
962:
958:
951:
948:
942:
937:
933:
926:
921:
917:
910:
908:
903:
892:
891:
890:
888:
880:
862:
858:
854:
850:
846:
842:
841:
832:
828:
824:
823:
814:
810:
807:
803:
799:
795:
791:
788:
784:
780:
779:
770:
765:
761:
757:
753:
749:
745:
744:conic section
741:
738:
737:
728:
725:
721:
717:
714:
713:area centroid
710:
707:
703:
699:
695:
691:
688:
685:are given in
684:
680:
679:
675:
669:
663:
645:
642:
638:
620:
605:
604:quadrilateral
601:
600:
591:
587:
583:
579:
575:
572:
571:Lemoine point
568:
564:
561:
560:
543:
538:
534:
528:
524:
520:
515:
511:
505:
501:
497:
492:
488:
482:
478:
474:
469:
465:
459:
455:
451:
446:
442:
436:
432:
428:
423:
419:
413:
409:
401:
400:
399:
398:
380:
376:
372:
367:
363:
359:
354:
350:
341:
323:
319:
315:
310:
306:
302:
297:
293:
284:
281:
278:
274:
270:
267:A triangle's
266:
263:
259:
256:
252:
248:
245:
241:
238:
234:
230:
226:
222:
218:
214:
210:
206:
202:
199:
195:
191:
187:
184:
180:
177:
173:
169:
165:
161:
157:
153:
149:
148:
147:
134:
132:
131:
126:
122:
121:vector spaces
118:
114:
113:
107:
105:
101:
96:
92:
88:
84:
80:
70:
68:
64:
60:
56:
52:
48:
41:
37:
33:
19:
3056:
3033:
3012:
2990:
2985:
2960:
2956:
2943:
2931:
2924:
2906:
2893:
2883:
2879:
2866:
2858:
2853:
2843:
2838:
2829:
2820:
2812:
2793:
2784:, pg. 178),
2772:
2714:through the
2698:, to relate
2685:
2657:
2634:
2632:
2602:
2431:
2269:
2265:
2260:
2258:
2248:
2241:
2232:
2227:
2222:
2217:
2213:
2071:
2062:
2057:
2050:collinearity
2049:
2043:
2007:
1998:
1984:
1974:
1962:
1950:
1946:
1942:
1938:
1919:Two numbers
1918:
1906:
1902:
1898:
1894:
1890:
1886:
1876:
1872:
1865:
1861:
1854:
1850:
1843:
1832:
1828:
1824:
1820:
1816:
1812:
1805:
1543:
1539:
1524:
1504:
1497:
1311:
1304:
1137:
876:
852:
849:circumcenter
838:Tetrahedrons
698:circumcenter
667:
661:
602:In a convex
251:circumcircle
164:Exeter point
156:circumcenter
145:
130:homographies
128:
112:collineation
110:
108:
103:
78:
76:
66:
62:
53:of a set of
51:collinearity
50:
44:
36:colinear map
2738:Coplanarity
2700:coordinates
2682:Photography
1985:concurrency
1975:In various
1511:determinant
845:Monge point
831:conic solid
637:Newton line
582:orthocenter
255:Simson line
207:), and the
194:Nagel point
152:orthocenter
3001:References
2136:, we have
2046:statistics
857:Euler line
853:Euler line
706:anticenter
569:, and the
229:its center
192:, and the
176:Euler line
2886:: 289–295
2788:, pg.106)
2612:ε
2582:ε
2561:−
2542:−
2535:λ
2528:⋯
2503:λ
2477:λ
2464:λ
2403:−
2384:−
2377:λ
2370:⋯
2345:λ
2319:λ
2306:λ
2177:λ
2164:λ
2114:λ
2087:λ
1937:(0, 0), (
1556:, etc.):
1463:…
1415:…
1367:…
1316:, if the
1270:…
1227:…
1184:…
1097:…
1023:…
949:…
813:hyperbola
756:hyperbola
619:midpoints
521:⋅
498:⋅
452:⋅
429:⋅
262:altitudes
237:perimeter
142:Triangles
79:collinear
63:collinear
18:Collinear
3091:Category
3053:(1968),
3032:(1969),
3013:Geometry
3011:(1998),
2901:(1995),
2727:See also
2259:Perfect
2024:adjacent
1999:Given a
1927:are not
1527:straight
802:vertices
752:parabola
734:Hexagons
724:incircle
672:. (See
340:extended
277:excircle
273:altitude
269:incenter
221:incircle
198:splitter
190:excircle
160:centroid
104:in a row
67:colinear
47:geometry
3081:0233275
2977:3677154
2718:of the
2676:antenna
1949:), (0,
1941:, 0), (
1929:coprime
1846:A, B, C
1839:A, B, C
1531:A, B, C
1314:X, Y, Z
1138:if the
868:Algebra
794:ellipse
787:circles
760:hexagon
748:ellipse
746:(i.e.,
235:of the
231:is the
223:of the
219:is the
213:cleaver
32:synteny
3079:
3069:
3040:
3019:
2975:
2913:
2720:camera
2712:object
2704:sensor
2666:is an
1957:(0, 0)
1881:, the
1498:is of
1318:matrix
1305:is of
1140:matrix
1103:
1094:
1078:
1029:
1020:
1004:
955:
946:
930:
792:In an
700:, the
696:, the
617:, the
227:, and
166:, the
162:, the
158:, the
154:, the
55:points
2973:S2CID
2953:(PDF)
2936:(PDF)
2876:(PDF)
2764:Notes
2668:array
2016:graph
2014:is a
1848:with
881:, in
829:of a
820:Cones
811:In a
718:In a
692:In a
95:model
81:. In
3067:ISBN
3038:ISBN
3017:ISBN
2911:ISBN
2694:and
2686:The
2662:, a
2272:≥ 2)
2105:and
2069:and
1963:m, n
1959:and
1923:and
1901:) +
1893:) ≤
1870:and
1552:and
1500:rank
1307:rank
887:rank
847:and
825:The
798:foci
613:and
607:ABCD
576:Two
150:The
87:line
59:line
2965:doi
2670:of
2658:In
2044:In
2010:of
1827:),
1819:),
1569:det
1507:= 2
877:In
781:By
754:or
665:= 2
621:of
119:of
45:In
3093::
3077:MR
3075:,
3061:,
2971:.
2961:13
2959:.
2955:.
2905:,
2882:,
2878:,
2827:.
2804:^
2247:,
2226:,
2048:,
2030:.
1967:.
1945:,
1907:BC
1899:AB
1891:AC
1877:BC
1866:AB
1855:AC
1833:AC
1825:BC
1817:AB
1789:0.
1544:AB
750:,
676:.)
668:GO
662:HG
632:EF
630:,
628:BD
626:,
624:AC
106:.
49:,
2979:.
2967::
2888:.
2884:6
2616:i
2586:i
2578:+
2573:i
2570:,
2567:)
2564:1
2558:k
2555:(
2551:X
2545:1
2539:k
2531:+
2525:+
2520:i
2517:2
2513:X
2507:2
2499:+
2494:i
2491:1
2487:X
2481:1
2473:+
2468:0
2460:=
2455:i
2452:k
2448:X
2434:i
2415:i
2412:,
2409:)
2406:1
2400:k
2397:(
2393:X
2387:1
2381:k
2373:+
2367:+
2362:i
2359:2
2355:X
2349:2
2341:+
2336:i
2333:1
2329:X
2323:1
2315:+
2310:0
2302:=
2297:i
2294:k
2290:X
2270:k
2268:(
2266:k
2254:)
2252:2
2249:X
2245:1
2242:X
2240:(
2236:)
2233:i
2231:2
2228:X
2223:i
2221:1
2218:X
2216:(
2199:.
2194:i
2191:1
2187:X
2181:1
2173:+
2168:0
2160:=
2155:i
2152:2
2148:X
2134:i
2118:1
2091:0
2075:2
2072:X
2066:1
2063:X
2028:P
2020:P
2012:P
2004:P
1965:)
1961:(
1953:)
1951:n
1947:n
1943:m
1939:m
1925:n
1921:m
1909:)
1905:(
1903:d
1897:(
1895:d
1889:(
1887:d
1879:)
1875:(
1873:d
1868:)
1864:(
1862:d
1857:)
1853:(
1851:d
1835:)
1831:(
1829:d
1823:(
1821:d
1815:(
1813:d
1786:=
1781:]
1775:0
1770:1
1765:1
1760:1
1753:1
1748:0
1741:2
1737:)
1733:C
1730:B
1727:(
1724:d
1717:2
1713:)
1709:C
1706:A
1703:(
1700:d
1693:1
1686:2
1682:)
1678:C
1675:B
1672:(
1669:d
1664:0
1657:2
1653:)
1649:B
1646:A
1643:(
1640:d
1633:1
1626:2
1622:)
1618:C
1615:A
1612:(
1609:d
1602:2
1598:)
1594:B
1591:A
1588:(
1585:d
1580:0
1574:[
1554:B
1550:A
1546:)
1542:(
1540:d
1505:n
1481:]
1473:n
1469:z
1456:2
1452:z
1444:1
1440:z
1434:1
1425:n
1421:y
1408:2
1404:y
1396:1
1392:y
1386:1
1377:n
1373:x
1360:2
1356:x
1348:1
1344:x
1338:1
1332:[
1288:]
1280:n
1276:z
1263:2
1259:z
1251:1
1247:z
1237:n
1233:y
1220:2
1216:y
1208:1
1204:y
1194:n
1190:x
1177:2
1173:x
1165:1
1161:x
1154:[
1119:,
1116:)
1111:n
1107:z
1100:,
1091:,
1086:2
1082:z
1075:,
1070:1
1066:z
1062:(
1059:=
1052:Z
1045:,
1042:)
1037:n
1033:y
1026:,
1017:,
1012:2
1008:y
1001:,
996:1
992:y
988:(
985:=
978:Y
971:,
968:)
963:n
959:x
952:,
943:,
938:2
934:x
927:,
922:1
918:x
914:(
911:=
904:X
883:n
771:.
689:.
656:O
652:G
648:H
615:F
611:E
592:.
544:.
539:2
535:A
529:3
525:P
516:1
512:A
506:2
502:P
493:3
489:A
483:1
479:P
475:=
470:1
466:A
460:3
456:P
447:3
443:A
437:2
433:P
424:2
420:A
414:1
410:P
381:3
377:A
373:,
368:2
364:A
360:,
355:1
351:A
324:3
320:P
316:,
311:2
307:P
303:,
298:1
294:P
185:.
178:.
42:.
20:)
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