56:
38:
2612:
985:
1357:
1642:
1503:
462:
260:
2270:
2059:
769:
2476:
1773:
780:
1219:
1108:
2447:
1230:
2364:
1525:
1386:
345:
146:
2156:
1945:
614:
2130:
1934:
1869:
2607:{\displaystyle {\frac {\overline {BD}}{\overline {DC}}}={\frac {\lambda _{C}}{\lambda _{B}}}\quad {\text{and}}\quad {\frac {\overline {CE}}{\overline {EA}}}={\frac {\lambda _{A}}{\lambda _{C}}}.}
2685:)-face. Each of these points divides the face on which it lies into lobes. Given a cycle of pairs of lobes, the product of the ratios of the volumes of the lobes in each pair is 1.
2691:
gives the area of the triangle formed by three cevians in the case that they are not concurrent. Ceva's theorem can be obtained from it by setting the area equal to zero and solving.
2705:
The theorem also has a well-known generalization to spherical and hyperbolic geometry, replacing the lengths in the ratios with their sines and hyperbolic sines, respectively.
980:{\displaystyle {\frac {\overline {BD}}{\overline {DC}}}={\frac {|\triangle BAD|-|\triangle BOD|}{|\triangle CAD|-|\triangle COD|}}={\frac {|\triangle ABO|}{|\triangle CAO|}}.}
1119:
1008:
1688:
562:
The first one is very elementary, using only basic properties of triangle areas. However, several cases have to be considered, depending on the position of the point
2375:
1352:{\displaystyle \left|{\frac {\overline {AF}}{\overline {FB}}}\cdot {\frac {\overline {BD}}{\overline {DC}}}\cdot {\frac {\overline {CE}}{\overline {EA}}}\right|=1,}
2293:
2663:
extends the conclusion of Ceva's theorem that the product of certain ratios is 1. Starting from a point in a simplex, a point is defined inductively on each
1637:{\displaystyle {\frac {\overline {BA}}{\overline {AF}}}\cdot {\frac {\overline {FO}}{\overline {OC}}}\cdot {\frac {\overline {CD}}{\overline {DB}}}=-1.}
1498:{\displaystyle {\frac {\overline {AB}}{\overline {BF}}}\cdot {\frac {\overline {FO}}{\overline {OC}}}\cdot {\frac {\overline {CE}}{\overline {EA}}}=-1}
457:{\displaystyle {\frac {\overline {AF}}{\overline {FB}}}\cdot {\frac {\overline {BD}}{\overline {DC}}}\cdot {\frac {\overline {CE}}{\overline {EA}}}=1,}
255:{\displaystyle {\frac {\overline {AF}}{\overline {FB}}}\cdot {\frac {\overline {BD}}{\overline {DC}}}\cdot {\frac {\overline {CE}}{\overline {EA}}}=1.}
308:, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two
2265:{\displaystyle {\overrightarrow {FO}}-\lambda _{C}{\overrightarrow {FC}}=\lambda _{A}{\overrightarrow {FA}}+\lambda _{B}{\overrightarrow {FB}}.}
2054:{\displaystyle {\overrightarrow {XO}}=\lambda _{A}{\overrightarrow {XA}}+\lambda _{B}{\overrightarrow {XB}}+\lambda _{C}{\overrightarrow {XC}},}
764:{\displaystyle {\frac {|\triangle BOD|}{|\triangle COD|}}={\frac {\overline {BD}}{\overline {DC}}}={\frac {|\triangle BAD|}{|\triangle CAD|}}.}
2782:
2630:
3120:
2698:
in the plane has been known since the early nineteenth century. The theorem has also been generalized to triangles on other surfaces of
3210:
3205:
3163:
Experimentally finding the centroid of a triangle with different weights at the vertices: a practical application of Ceva's theorem
3096:
2082:
1877:
3137:
1821:
3183:
548:
491:
3178:
3200:
608:
To check the magnitude, note that the area of a triangle of a given height is proportional to its base. So
3123:
includes definitions of cevian triangle, cevian nest, anticevian triangle, Ceva conjugate, and cevapoint
2845:
2725:
31:
2730:
1366:
2079:
is supposed to not belong to any line passing through two vertices of the triangle. This implies that
2740:
540:
1214:{\displaystyle {\frac {\overline {AF}}{\overline {FB}}}={\frac {|\triangle CAO|}{|\triangle BCO|}}.}
1103:{\displaystyle {\frac {\overline {CE}}{\overline {EA}}}={\frac {|\triangle BCO|}{|\triangle ABO|}},}
601:
is inside the triangle (upper diagram), or one is positive and the other two are negative, the case
3162:
2714:
2645:
1768:{\displaystyle {\frac {\overline {AF}}{\overline {FB}}}={\frac {\overline {AF'}}{\overline {F'B}}}}
582:
476:
321:
3173:
3041:
3006:
2971:
2932:
2897:
2699:
74:
2688:
2774:
2287:
are not collinear. It follows that the two members of the equation equal the zero vector, and
3145:
2778:
2719:
2649:
2442:{\displaystyle {\frac {\overline {AF}}{\overline {FB}}}={\frac {\lambda _{B}}{\lambda _{A}}},}
574:
100:
3076:
3033:
2998:
2963:
2924:
2889:
2766:
1803:
472:
2799:
2652:
can be assigned to the vertices such that each cevian intersects the opposite facet at its
2880:
Landy, Steven (December 1988). "A Generalization of Ceva's
Theorem to Higher Dimensions".
305:
93:
27:
Geometric relation between line segments from a triangle's vertices and their intersection
3166:
2915:
Wernicke, Paul (November 1927). "The
Theorems of Ceva and Menelaus and Their Extension".
2864:
3102:
2656:. Moreover, the intersection point of the cevians is the center of mass of the simplex.
2359:{\displaystyle \lambda _{A}{\overrightarrow {FA}}+\lambda _{B}{\overrightarrow {FB}}=0.}
2275:
The left-hand side of this equation is a vector that has the same direction as the line
559:
Several proofs of the theorem have been given. Two proofs are given in the following.
2653:
594:
3126:
3111:
2617:
Ceva's theorem results immediately by taking the product of the three last equations.
2452:
where the left-hand-side fraction is the signed ratio of the lengths of the collinear
3194:
3045:
2975:
2767:
483:
3169:, an interactive dynamic geometry sketch using the gravity simulator of Cinderella.
3115:
3106:
2989:
Grünbaum, Branko; Shephard, G. C. (1995). "Ceva, Menelaus and the Area
Principle".
2453:
2069:
1807:
1795:
578:
570:
317:
313:
309:
137:
3148:
3132:
2967:
55:
37:
3065:"Al-Mutaman ibn Hűd, 11the century king of Saragossa and brilliant mathematician"
2830:
2667:-face. This point is the foot of a cevian that goes from the vertex opposite the
3024:
Masal'tsev, L. A. (1994). "Incidence theorems in spaces of constant curvature".
577:, but is somehow more natural and not case dependent. Moreover, it works in any
544:
17:
597:
is positive since either all three of the ratios are positive, the case where
3153:
543:
in that their equations differ only in sign. By re-writing each in terms of
3081:
3064:
2800:"A Unified Proof of Ceva and Menelaus' Theorems Using Projective Geometry"
2735:
495:
82:
501:
Associated with the figures are several terms derived from Ceva's name:
3037:
3010:
2936:
2901:
2695:
2660:
2626:
328:
2745:
502:
133:
3002:
2928:
2893:
2678:)-face that contains it, through the point already defined on this (
2068:(for the definition of this arrow notation and further details, see
59:
Ceva's theorem, case 2: the three lines are concurrent at a point
54:
41:
Ceva's theorem, case 1: the three lines are concurrent at a point
36:
2637:-simplex as a ray from each vertex to a point on the opposite (
2852:, pages 177–180, Dover Publishing Co., second revised edition.
2279:, and the right-hand side has the same direction as the line
1778:
But at most one point can cut a segment in a given ratio so
2829:
Russell, John
Wellesley (1905). "Ch. 1 §7 Ceva's Theorem".
2125:{\displaystyle \lambda _{A}\lambda _{B}\lambda _{C}\neq 0.}
2950:
Samet, Dov (May 2021). "An
Extension of Ceva's Theorem to
3127:
Conics
Associated with a Cevian Nest, by Clark Kimberling
1929:{\displaystyle \lambda _{A}+\lambda _{B}+\lambda _{C}=1,}
479:. The converse is often included as part of the theorem.
271:
is taken to be positive or negative according to whether
2150:(see figures), the last equation may be rearranged into
605:
is outside the triangle (lower diagram shows one case).
528:); cevian nest, anticevian triangle, Ceva conjugate. (
1864:{\displaystyle \lambda _{A},\lambda _{B},\lambda _{C}}
2648:). Then the cevians are concurrent if and only if a
2625:
The theorem can be generalized to higher-dimensional
2479:
2378:
2296:
2159:
2085:
1948:
1880:
1824:
1691:
1647:
The theorem follows by dividing these two equations.
1528:
1389:
1233:
1122:
1011:
783:
617:
348:
149:
279:
in some fixed orientation of the line. For example,
1678:. Then by the theorem, the equation also holds for
2606:
2441:
2358:
2264:
2124:
2053:
1928:
1863:
1767:
1636:
1497:
1351:
1213:
1102:
979:
763:
456:
254:
3103:Derivations and applications of Ceva's Theorem
2283:. These lines have different directions since
2659:Another generalization to higher-dimensional
316:). It is therefore true for triangles in any
8:
3121:Glossary of Encyclopedia of Triangle Centers
1365:The theorem can also be proven easily using
2863:Hopkins, George Irving (1902). "Art. 986".
3080:
2593:
2583:
2577:
2544:
2538:
2529:
2519:
2513:
2480:
2478:
2428:
2418:
2412:
2379:
2377:
2335:
2329:
2307:
2301:
2295:
2244:
2238:
2216:
2210:
2188:
2182:
2160:
2158:
2110:
2100:
2090:
2084:
2033:
2027:
2005:
1999:
1977:
1971:
1949:
1947:
1911:
1898:
1885:
1879:
1855:
1842:
1829:
1823:
1725:
1692:
1690:
1650:The converse follows as a corollary. Let
1595:
1562:
1529:
1527:
1456:
1423:
1390:
1388:
1305:
1272:
1239:
1232:
1200:
1183:
1176:
1159:
1156:
1123:
1121:
1089:
1072:
1065:
1048:
1045:
1012:
1010:
966:
949:
942:
925:
922:
911:
894:
886:
869:
862:
845:
837:
820:
817:
784:
782:
750:
733:
726:
709:
706:
673:
662:
645:
638:
621:
618:
616:
415:
382:
349:
347:
289:is defined as having positive value when
216:
183:
150:
148:
2694:The analogue of the theorem for general
1224:Multiplying these three equations gives
2757:
7:
3112:Trigonometric Form of Ceva's Theorem
2824:
2822:
2820:
490:. But it was proven much earlier by
486:, who published it in his 1678 work
482:The theorem is often attributed to
1188:
1164:
1077:
1053:
990:(Replace the minus with a plus if
954:
930:
899:
874:
850:
825:
738:
714:
650:
626:
547:, the two theorems may be seen as
25:
2956:The American Mathematical Monthly
2917:The American Mathematical Monthly
2882:The American Mathematical Monthly
2807:Journal for Geometry and Graphics
3026:Journal of Mathematical Sciences
2850:Challenging Problems in Geometry
1658:so that the equation holds. Let
2848:and Charles T. Salkind (1996),
2769:Geometry: Our Cultural Heritage
2543:
2537:
539:The theorem is very similar to
304:Ceva's theorem is a theorem of
3138:Wolfram Demonstrations Project
2731:Menelaus's theorem - Knowledge
2075:For Ceva's theorem, the point
1201:
1184:
1177:
1160:
1090:
1073:
1066:
1049:
967:
950:
943:
926:
912:
895:
887:
870:
863:
846:
838:
821:
751:
734:
727:
710:
663:
646:
639:
622:
494:, an eleventh-century king of
1:
2968:10.1080/00029890.2021.1896292
2741:Stewart's theorem - Knowledge
1818:are the unique three numbers
1786:Using barycentric coordinates
114:), to meet opposite sides at
2568:
2555:
2504:
2491:
2403:
2390:
1759:
1741:
1716:
1703:
1619:
1606:
1586:
1573:
1553:
1540:
1480:
1467:
1447:
1434:
1414:
1401:
1329:
1316:
1296:
1283:
1263:
1250:
1147:
1134:
1036:
1023:
808:
795:
697:
684:
439:
426:
406:
393:
373:
360:
240:
227:
207:
194:
174:
161:
118:respectively. (The segments
107:(not on one of the sides of
3179:Encyclopedia of Mathematics
1802:, that belongs to the same
275:is to the left or right of
265:In other words, the length
3227:
524:is the cevian triangle of
138:signed lengths of segments
29:
3167:Dynamic Geometry Sketches
3063:Hogendijk, J. B. (1995).
2470:The same reasoning shows
1508:and from the transversal
998:are on opposite sides of
536:is pronounced chev'ian.)
492:Yusuf Al-Mu'taman ibn Hűd
3211:Euclidean plane geometry
3206:Theorems about triangles
2866:Inductive Plane Geometry
2633:. Define a cevian of an
331:is also true: If points
301:and negative otherwise.
2798:Benitez, Julio (2007).
2631:barycentric coordinates
1808:barycentric coordinates
1369:. From the transversal
593:First, the sign of the
571:barycentric coordinates
532:is pronounced Chay'va;
3082:10.1006/hmat.1995.1001
2869:. D.C. Heath & Co.
2608:
2443:
2360:
2266:
2126:
2055:
1930:
1865:
1769:
1682:. Comparing the two,
1654:be given on the lines
1638:
1499:
1353:
1215:
1104:
981:
765:
569:The second proof uses
458:
339:respectively so that
256:
70:
52:
2846:Alfred S. Posamentier
2765:Holme, Audun (2010).
2726:Circumcevian triangle
2609:
2444:
2361:
2267:
2127:
2056:
1931:
1866:
1770:
1639:
1500:
1354:
1216:
1105:
982:
766:
459:
257:
58:
40:
32:Ceva (disambiguation)
3069:Historia Mathematica
2991:Mathematics Magazine
2773:. Springer. p.
2736:Triangle - Knowledge
2477:
2376:
2294:
2157:
2083:
1946:
1878:
1822:
1689:
1526:
1387:
1231:
1120:
1009:
781:
615:
589:Using triangle areas
346:
147:
30:For other uses, see
3136:by Jay Warendorff,
2715:Projective geometry
1790:Given three points
1670:be the point where
509:are the cevians of
327:A slightly adapted
85:. Given a triangle
81:is a theorem about
3146:Weisstein, Eric W.
3038:10.1007/BF01249519
2835:. Clarendon Press.
2746:Cevian - Knowledge
2700:constant curvature
2604:
2439:
2356:
2262:
2122:
2051:
1926:
1861:
1765:
1634:
1495:
1367:Menelaus's theorem
1349:
1211:
1100:
977:
761:
454:
252:
103:to a common point
99:be drawn from the
75:Euclidean geometry
71:
53:
3097:Menelaus and Ceva
2784:978-3-642-14440-0
2720:Median (geometry)
2650:mass distribution
2599:
2572:
2571:
2558:
2541:
2535:
2508:
2507:
2494:
2434:
2407:
2406:
2393:
2369:It follows that
2348:
2320:
2257:
2229:
2201:
2173:
2138:the intersection
2134:If one takes for
2046:
2018:
1990:
1962:
1763:
1762:
1744:
1720:
1719:
1706:
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1622:
1609:
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1483:
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1437:
1418:
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1404:
1333:
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1300:
1299:
1286:
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1266:
1253:
1206:
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1137:
1095:
1040:
1039:
1026:
972:
917:
812:
811:
798:
756:
701:
700:
687:
668:
541:Menelaus' theorem
443:
442:
429:
410:
409:
396:
377:
376:
363:
244:
243:
230:
211:
210:
197:
178:
177:
164:
16:(Redirected from
3218:
3187:
3159:
3158:
3149:"Ceva's Theorem"
3086:
3084:
3050:
3049:
3032:(4): 3201–3206.
3021:
3015:
3014:
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2979:
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2940:
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2906:
2905:
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2837:
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2826:
2815:
2814:
2804:
2795:
2789:
2788:
2772:
2762:
2722:– an application
2684:
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2131:
2129:
2128:
2123:
2115:
2114:
2105:
2104:
2095:
2094:
2078:
2067:
2064:for every point
2060:
2058:
2057:
2052:
2047:
2042:
2034:
2032:
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2019:
2014:
2006:
2004:
2003:
1991:
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1916:
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1870:
1868:
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1862:
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1834:
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1817:
1814:with respect of
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1801:
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707:
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696:
688:
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675:
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669:
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666:
649:
643:
642:
625:
619:
604:
600:
565:
549:projective duals
527:
523:
512:
508:
488:De lineis rectis
470:
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460:
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51:
44:
21:
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3216:
3215:
3201:Affine geometry
3191:
3190:
3172:
3144:
3143:
3093:
3062:
3059:
3057:Further reading
3054:
3053:
3023:
3022:
3018:
3003:10.2307/2690569
2988:
2987:
2983:
2949:
2948:
2944:
2929:10.2307/2300222
2914:
2913:
2909:
2894:10.2307/2322390
2888:(10): 936–939.
2879:
2878:
2874:
2862:
2860:
2856:
2844:
2840:
2828:
2827:
2818:
2802:
2797:
2796:
2792:
2785:
2764:
2763:
2759:
2754:
2711:
2689:Routh's theorem
2679:
2672:
2668:
2664:
2638:
2634:
2623:
2621:Generalizations
2589:
2579:
2560:
2547:
2525:
2515:
2496:
2483:
2475:
2474:
2463:
2462:
2457:
2456:
2424:
2414:
2395:
2382:
2374:
2373:
2337:
2325:
2309:
2297:
2292:
2291:
2284:
2280:
2276:
2246:
2234:
2218:
2206:
2190:
2178:
2162:
2155:
2154:
2147:
2143:
2139:
2135:
2106:
2096:
2086:
2081:
2080:
2076:
2065:
2035:
2023:
2007:
1995:
1979:
1967:
1951:
1944:
1943:
1907:
1894:
1881:
1876:
1875:
1851:
1838:
1825:
1820:
1819:
1815:
1811:
1799:
1791:
1788:
1779:
1747:
1746:
1732:
1728:
1708:
1695:
1687:
1686:
1679:
1675:
1671:
1667:
1663:
1659:
1655:
1651:
1611:
1598:
1578:
1565:
1545:
1532:
1524:
1523:
1513:
1509:
1472:
1459:
1439:
1426:
1406:
1393:
1385:
1384:
1374:
1370:
1321:
1308:
1288:
1275:
1255:
1242:
1238:
1234:
1229:
1228:
1182:
1158:
1139:
1126:
1118:
1117:
1071:
1047:
1028:
1015:
1007:
1006:
999:
995:
991:
948:
924:
868:
819:
800:
787:
779:
778:
732:
708:
689:
676:
644:
620:
613:
612:
602:
598:
591:
563:
557:
525:
518:
515:cevian triangle
510:
506:
475:, or all three
468:
431:
418:
398:
385:
365:
352:
344:
343:
336:
332:
306:affine geometry
298:
294:
290:
285:
281:
280:
276:
272:
267:
266:
232:
219:
199:
186:
166:
153:
145:
144:
136:.) Then, using
128:
124:
120:
119:
115:
108:
104:
96:
86:
64:
60:
46:
42:
35:
28:
23:
22:
18:Cevian triangle
15:
12:
11:
5:
3224:
3222:
3214:
3213:
3208:
3203:
3193:
3192:
3189:
3188:
3174:"Ceva theorem"
3170:
3160:
3141:
3133:Ceva's Theorem
3129:
3124:
3118:
3109:
3100:
3092:
3091:External links
3089:
3088:
3087:
3058:
3055:
3052:
3051:
3016:
2997:(4): 254–268.
2981:
2962:(5): 435–445.
2942:
2923:(9): 468–472.
2907:
2872:
2854:
2838:
2816:
2790:
2783:
2756:
2755:
2753:
2750:
2749:
2748:
2743:
2738:
2733:
2728:
2723:
2717:
2710:
2707:
2654:center of mass
2622:
2619:
2615:
2614:
2603:
2596:
2592:
2586:
2582:
2576:
2570:
2566:
2563:
2557:
2553:
2550:
2532:
2528:
2522:
2518:
2512:
2506:
2502:
2499:
2493:
2489:
2486:
2450:
2449:
2438:
2431:
2427:
2421:
2417:
2411:
2405:
2401:
2398:
2392:
2388:
2385:
2367:
2366:
2355:
2352:
2347:
2343:
2340:
2332:
2328:
2324:
2319:
2315:
2312:
2304:
2300:
2273:
2272:
2261:
2256:
2252:
2249:
2241:
2237:
2233:
2228:
2224:
2221:
2213:
2209:
2205:
2200:
2196:
2193:
2185:
2181:
2177:
2172:
2168:
2165:
2121:
2118:
2113:
2109:
2103:
2099:
2093:
2089:
2062:
2061:
2050:
2045:
2041:
2038:
2030:
2026:
2022:
2017:
2013:
2010:
2002:
1998:
1994:
1989:
1985:
1982:
1974:
1970:
1966:
1961:
1957:
1954:
1937:
1936:
1925:
1922:
1919:
1914:
1910:
1906:
1901:
1897:
1893:
1888:
1884:
1858:
1854:
1850:
1845:
1841:
1837:
1832:
1828:
1798:, and a point
1787:
1784:
1776:
1775:
1761:
1757:
1753:
1750:
1743:
1738:
1735:
1731:
1724:
1718:
1714:
1711:
1705:
1701:
1698:
1645:
1644:
1633:
1630:
1627:
1621:
1617:
1614:
1608:
1604:
1601:
1594:
1588:
1584:
1581:
1575:
1571:
1568:
1561:
1555:
1551:
1548:
1542:
1538:
1535:
1506:
1505:
1494:
1491:
1488:
1482:
1478:
1475:
1469:
1465:
1462:
1455:
1449:
1445:
1442:
1436:
1432:
1429:
1422:
1416:
1412:
1409:
1403:
1399:
1396:
1360:
1359:
1348:
1345:
1342:
1338:
1331:
1327:
1324:
1318:
1314:
1311:
1304:
1298:
1294:
1291:
1285:
1281:
1278:
1271:
1265:
1261:
1258:
1252:
1248:
1245:
1237:
1222:
1221:
1210:
1203:
1199:
1196:
1193:
1190:
1186:
1179:
1175:
1172:
1169:
1166:
1162:
1155:
1149:
1145:
1142:
1136:
1132:
1129:
1111:
1110:
1099:
1092:
1088:
1085:
1082:
1079:
1075:
1068:
1064:
1061:
1058:
1055:
1051:
1044:
1038:
1034:
1031:
1025:
1021:
1018:
1002:.) Similarly,
988:
987:
976:
969:
965:
962:
959:
956:
952:
945:
941:
938:
935:
932:
928:
921:
914:
910:
907:
904:
901:
897:
893:
889:
885:
882:
879:
876:
872:
865:
861:
858:
855:
852:
848:
844:
840:
836:
833:
830:
827:
823:
816:
810:
806:
803:
797:
793:
790:
772:
771:
760:
753:
749:
746:
743:
740:
736:
729:
725:
722:
719:
716:
712:
705:
699:
695:
692:
686:
682:
679:
672:
665:
661:
658:
655:
652:
648:
641:
637:
634:
631:
628:
624:
595:left-hand side
590:
587:
556:
553:
517:(the triangle
465:
464:
453:
450:
447:
441:
437:
434:
428:
424:
421:
414:
408:
404:
401:
395:
391:
388:
381:
375:
371:
368:
362:
358:
355:
335:are chosen on
263:
262:
251:
248:
242:
238:
235:
229:
225:
222:
215:
209:
205:
202:
196:
192:
189:
182:
176:
172:
169:
163:
159:
156:
79:Ceva's theorem
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3223:
3212:
3209:
3207:
3204:
3202:
3199:
3198:
3196:
3185:
3181:
3180:
3175:
3171:
3168:
3164:
3161:
3156:
3155:
3150:
3147:
3142:
3139:
3135:
3134:
3130:
3128:
3125:
3122:
3119:
3117:
3113:
3110:
3108:
3104:
3101:
3098:
3095:
3094:
3090:
3083:
3078:
3074:
3070:
3066:
3061:
3060:
3056:
3047:
3043:
3039:
3035:
3031:
3027:
3020:
3017:
3012:
3008:
3004:
3000:
2996:
2992:
2985:
2982:
2977:
2973:
2969:
2965:
2961:
2957:
2954:-Simplices".
2953:
2946:
2943:
2938:
2934:
2930:
2926:
2922:
2918:
2911:
2908:
2903:
2899:
2895:
2891:
2887:
2883:
2876:
2873:
2868:
2867:
2858:
2855:
2851:
2847:
2842:
2839:
2834:
2833:
2832:Pure Geometry
2825:
2823:
2821:
2817:
2812:
2808:
2801:
2794:
2791:
2786:
2780:
2776:
2771:
2770:
2761:
2758:
2751:
2747:
2744:
2742:
2739:
2737:
2734:
2732:
2729:
2727:
2724:
2721:
2718:
2716:
2713:
2712:
2708:
2706:
2703:
2701:
2697:
2692:
2690:
2686:
2682:
2675:
2671:-face, in a (
2662:
2657:
2655:
2651:
2647:
2641:
2632:
2628:
2620:
2618:
2601:
2594:
2590:
2584:
2580:
2574:
2564:
2561:
2551:
2548:
2530:
2526:
2520:
2516:
2510:
2500:
2497:
2487:
2484:
2473:
2472:
2471:
2468:
2455:
2454:line segments
2436:
2429:
2425:
2419:
2415:
2409:
2399:
2396:
2386:
2383:
2372:
2371:
2370:
2353:
2350:
2345:
2341:
2338:
2330:
2326:
2322:
2317:
2313:
2310:
2302:
2298:
2290:
2289:
2288:
2259:
2254:
2250:
2247:
2239:
2235:
2231:
2226:
2222:
2219:
2211:
2207:
2203:
2198:
2194:
2191:
2183:
2179:
2175:
2170:
2166:
2163:
2153:
2152:
2151:
2142:of the lines
2132:
2119:
2116:
2111:
2107:
2101:
2097:
2091:
2087:
2073:
2071:
2048:
2043:
2039:
2036:
2028:
2024:
2020:
2015:
2011:
2008:
2000:
1996:
1992:
1987:
1983:
1980:
1972:
1968:
1964:
1959:
1955:
1952:
1942:
1941:
1940:
1923:
1920:
1917:
1912:
1908:
1904:
1899:
1895:
1891:
1886:
1882:
1874:
1873:
1872:
1856:
1852:
1848:
1843:
1839:
1835:
1830:
1826:
1809:
1805:
1797:
1794:that are not
1785:
1783:
1755:
1751:
1748:
1736:
1733:
1729:
1722:
1712:
1709:
1699:
1696:
1685:
1684:
1683:
1648:
1631:
1628:
1625:
1615:
1612:
1602:
1599:
1592:
1582:
1579:
1569:
1566:
1559:
1549:
1546:
1536:
1533:
1522:
1521:
1520:
1517:
1492:
1489:
1486:
1476:
1473:
1463:
1460:
1453:
1443:
1440:
1430:
1427:
1420:
1410:
1407:
1397:
1394:
1383:
1382:
1381:
1378:
1368:
1363:
1362:as required.
1346:
1343:
1340:
1336:
1325:
1322:
1312:
1309:
1302:
1292:
1289:
1279:
1276:
1269:
1259:
1256:
1246:
1243:
1235:
1227:
1226:
1225:
1208:
1197:
1194:
1191:
1173:
1170:
1167:
1153:
1143:
1140:
1130:
1127:
1116:
1115:
1114:
1097:
1086:
1083:
1080:
1062:
1059:
1056:
1042:
1032:
1029:
1019:
1016:
1005:
1004:
1003:
974:
963:
960:
957:
939:
936:
933:
919:
908:
905:
902:
891:
883:
880:
877:
859:
856:
853:
842:
834:
831:
828:
814:
804:
801:
791:
788:
777:
776:
775:
758:
747:
744:
741:
723:
720:
717:
703:
693:
690:
680:
677:
670:
659:
656:
653:
635:
632:
629:
611:
610:
609:
606:
596:
588:
586:
584:
580:
576:
572:
567:
560:
554:
552:
550:
546:
542:
537:
535:
531:
522:
516:
504:
499:
497:
493:
489:
485:
484:Giovanni Ceva
480:
478:
474:
451:
448:
445:
435:
432:
422:
419:
412:
402:
399:
389:
386:
379:
369:
366:
356:
353:
342:
341:
340:
330:
325:
323:
319:
315:
311:
310:line segments
307:
302:
249:
246:
236:
233:
223:
220:
213:
203:
200:
190:
187:
180:
170:
167:
157:
154:
143:
142:
141:
139:
135:
132:are known as
112:
102:
95:
90:
84:
80:
76:
68:
57:
50:
39:
33:
19:
3177:
3152:
3131:
3116:cut-the-knot
3107:cut-the-knot
3099:at MathPages
3072:
3068:
3029:
3025:
3019:
2994:
2990:
2984:
2959:
2955:
2951:
2945:
2920:
2916:
2910:
2885:
2881:
2875:
2865:
2857:
2849:
2841:
2831:
2810:
2806:
2793:
2768:
2760:
2704:
2693:
2687:
2680:
2673:
2658:
2639:
2624:
2616:
2469:
2451:
2368:
2274:
2133:
2074:
2070:Affine space
2063:
1938:
1789:
1777:
1649:
1646:
1515:
1512:of triangle
1507:
1376:
1373:of triangle
1364:
1361:
1223:
1112:
989:
773:
607:
592:
579:affine plane
568:
561:
558:
545:cross-ratios
538:
533:
529:
520:
514:
500:
487:
481:
466:
326:
318:affine plane
303:
264:
110:
88:
78:
72:
66:
48:
2813:(1): 39–44.
774:Therefore,
505:(the lines
293:is between
3195:Categories
2752:References
1871:such that
1656:BC, AC, AB
507:AD, BE, CF
473:concurrent
469:AD, BE, CF
337:BC, AC, AB
97:AO, BO, CO
92:, let the
3184:EMS Press
3154:MathWorld
3046:123870381
2976:233413469
2661:simplexes
2627:simplexes
2591:λ
2581:λ
2569:¯
2556:¯
2527:λ
2517:λ
2505:¯
2492:¯
2426:λ
2416:λ
2404:¯
2391:¯
2346:→
2327:λ
2318:→
2299:λ
2255:→
2236:λ
2227:→
2208:λ
2199:→
2180:λ
2176:−
2171:→
2117:≠
2108:λ
2098:λ
2088:λ
2044:→
2025:λ
2016:→
1997:λ
1988:→
1969:λ
1960:→
1909:λ
1896:λ
1883:λ
1853:λ
1840:λ
1827:λ
1796:collinear
1760:¯
1742:¯
1717:¯
1704:¯
1629:−
1620:¯
1607:¯
1593:⋅
1587:¯
1574:¯
1560:⋅
1554:¯
1541:¯
1490:−
1481:¯
1468:¯
1454:⋅
1448:¯
1435:¯
1421:⋅
1415:¯
1402:¯
1330:¯
1317:¯
1303:⋅
1297:¯
1284:¯
1270:⋅
1264:¯
1251:¯
1189:△
1165:△
1148:¯
1135:¯
1078:△
1054:△
1037:¯
1024:¯
955:△
931:△
900:△
892:−
875:△
851:△
843:−
826:△
809:¯
796:¯
739:△
715:△
698:¯
685:¯
651:△
627:△
581:over any
440:¯
427:¯
413:⋅
407:¯
394:¯
380:⋅
374:¯
361:¯
320:over any
314:collinear
312:that are
241:¯
228:¯
214:⋅
208:¯
195:¯
181:⋅
175:¯
162:¯
83:triangles
3075:: 1–18.
2861:Follows
2709:See also
2696:polygons
2644:)-face (
1752:′
1737:′
1680:D, E, F'
1674:crosses
1666:and let
1662:meet at
496:Zaragoza
477:parallel
329:converse
101:vertices
63:outside
3186:, 2001
3011:2690569
2937:2300222
2902:2322390
2285:A, B, C
1816:A, B, C
1792:A, B, C
1652:D, E, F
575:vectors
333:D, E, F
134:cevians
116:D, E, F
45:inside
3044:
3009:
2974:
2935:
2900:
2781:
2629:using
1806:, the
1780:F = F’
1660:AD, BE
555:Proofs
534:cevian
503:cevian
3042:S2CID
3007:JSTOR
2972:S2CID
2933:JSTOR
2898:JSTOR
2803:(PDF)
2646:facet
1804:plane
583:field
467:then
322:field
94:lines
2779:ISBN
2461:and
2146:and
1939:and
1113:and
994:and
573:and
530:Ceva
471:are
297:and
3165:at
3114:at
3105:at
3077:doi
3034:doi
2999:doi
2964:doi
2960:128
2925:doi
2890:doi
2775:210
2683:+ 1
2676:+ 1
2642:– 1
2540:and
2072:).
1810:of
1516:BCF
1510:AOD
1377:ACF
1371:BOE
566:.
521:DEF
513:),
111:ABC
89:ABC
73:In
67:ABC
49:ABC
3197::
3182:,
3176:,
3151:.
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