45:
27:
2601:
974:
1346:
1631:
1492:
451:
249:
2259:
2048:
758:
2465:
1762:
769:
1208:
1097:
2436:
1219:
2353:
1514:
1375:
334:
135:
2145:
1934:
603:
2119:
1923:
1858:
2596:{\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}}}.}
2674:)-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.
2680:
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.
2694:
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.
969:{\displaystyle {\frac {\overline {BD}}{\overline {DC}}}={\frac {|\triangle BAD|-|\triangle BOD|}{|\triangle CAD|-|\triangle COD|}}={\frac {|\triangle ABO|}{|\triangle CAO|}}.}
1108:
997:
1677:
551:
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
2364:
1341:{\displaystyle \left|{\frac {\overline {AF}}{\overline {FB}}}\cdot {\frac {\overline {BD}}{\overline {DC}}}\cdot {\frac {\overline {CE}}{\overline {EA}}}\right|=1,}
2282:
2652:
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
1626:{\displaystyle {\frac {\overline {BA}}{\overline {AF}}}\cdot {\frac {\overline {FO}}{\overline {OC}}}\cdot {\frac {\overline {CD}}{\overline {DB}}}=-1.}
1487:{\displaystyle {\frac {\overline {AB}}{\overline {BF}}}\cdot {\frac {\overline {FO}}{\overline {OC}}}\cdot {\frac {\overline {CE}}{\overline {EA}}}=-1}
446:{\displaystyle {\frac {\overline {AF}}{\overline {FB}}}\cdot {\frac {\overline {BD}}{\overline {DC}}}\cdot {\frac {\overline {CE}}{\overline {EA}}}=1,}
244:{\displaystyle {\frac {\overline {AF}}{\overline {FB}}}\cdot {\frac {\overline {BD}}{\overline {DC}}}\cdot {\frac {\overline {CE}}{\overline {EA}}}=1.}
297:, 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
2254:{\displaystyle {\overrightarrow {FO}}-\lambda _{C}{\overrightarrow {FC}}=\lambda _{A}{\overrightarrow {FA}}+\lambda _{B}{\overrightarrow {FB}}.}
2043:{\displaystyle {\overrightarrow {XO}}=\lambda _{A}{\overrightarrow {XA}}+\lambda _{B}{\overrightarrow {XB}}+\lambda _{C}{\overrightarrow {XC}},}
753:{\displaystyle {\frac {|\triangle BOD|}{|\triangle COD|}}={\frac {\overline {BD}}{\overline {DC}}}={\frac {|\triangle BAD|}{|\triangle CAD|}}.}
3109:
2687:
in the plane has been known since the early nineteenth century. The theorem has also been generalized to triangles on other surfaces of
3152:
Experimentally finding the centroid of a triangle with different weights at the vertices: a practical application of Ceva's theorem
3085:
2071:
1866:
2771:
2619:
1810:
3199:
3194:
3126:
3172:
537:
597:
To check the magnitude, note that the area of a triangle of a given height is proportional to its base. So
480:
3167:
3112:
includes definitions of cevian triangle, cevian nest, anticevian triangle, Ceva conjugate, and cevapoint
2068:
is supposed to not belong to any line passing through two vertices of the triangle. This implies that
1203:{\displaystyle {\frac {\overline {AF}}{\overline {FB}}}={\frac {|\triangle CAO|}{|\triangle BCO|}}.}
1092:{\displaystyle {\frac {\overline {CE}}{\overline {EA}}}={\frac {|\triangle BCO|}{|\triangle ABO|}},}
590:
is inside the triangle (upper diagram), or one is positive and the other two are negative, the case
3189:
3151:
1796:
1757:{\displaystyle {\frac {\overline {AF}}{\overline {FB}}}={\frac {\overline {AF'}}{\overline {F'B}}}}
559:
3162:
126:
2763:
2276:
are not collinear. It follows that the two members of the equation equal the zero vector, and
2431:{\displaystyle {\frac {\overline {AF}}{\overline {FB}}}={\frac {\lambda _{B}}{\lambda _{A}}},}
2834:
2719:
2714:
1355:
20:
2755:
2788:
2729:
2641:
can be assigned to the vertices such that each cevian intersects the opposite facet at its
529:
2869:
Landy, Steven (December 1988). "A Generalization of Ceva's
Theorem to Higher Dimensions".
16:
Geometric relation between line segments from a triangle's vertices and their intersection
8:
3155:
2904:
Wernicke, Paul (November 1927). "The
Theorems of Ceva and Menelaus and Their Extension".
2853:
2703:
2634:
571:
465:
310:
3091:
2645:. Moreover, the intersection point of the cevians is the center of mass of the simplex.
2348:{\displaystyle \lambda _{A}{\overrightarrow {FA}}+\lambda _{B}{\overrightarrow {FB}}=0.}
2264:
The left-hand side of this equation is a vector that has the same direction as the line
548:
Several proofs of the theorem have been given. Two proofs are given in the following.
3030:
2995:
2960:
2921:
2886:
2688:
2677:
63:
3115:
3100:
2606:
Ceva's theorem results immediately by taking the product of the three last equations.
2441:
where the left-hand-side fraction is the signed ratio of the lengths of the collinear
3134:
3034:
2964:
2767:
2756:
2708:
2638:
563:
89:
3158:, an interactive dynamic geometry sketch using the gravity simulator of Cinderella.
3065:
3022:
2987:
2978:
Grünbaum, Branko; Shephard, G. C. (1995). "Ceva, Menelaus and the Area
Principle".
2952:
2913:
2878:
1792:
461:
3137:
3121:
2956:
3054:"Al-Mutaman ibn Hűd, 11the century king of Saragossa and brilliant mathematician"
2819:
2656:-face. This point is the foot of a cevian that goes from the vertex opposite the
294:
82:
3013:
Masal'tsev, L. A. (1994). "Incidence theorems in spaces of constant curvature".
566:, but is somehow more natural and not case dependent. Moreover, it works in any
2642:
583:
586:
is positive since either all three of the ratios are positive, the case where
3183:
472:
532:
in that their equations differ only in sign. By re-writing each in terms of
3104:
3095:
3070:
3053:
2442:
2058:
1784:
567:
306:
302:
298:
2789:"A Unified Proof of Ceva and Menelaus' Theorems Using Projective Geometry"
317:
533:
490:
Associated with the figures are several terms derived from Ceva's name:
3026:
2999:
2925:
2890:
3142:
44:
26:
2991:
2917:
2882:
2667:)-face that contains it, through the point already defined on this (
2057:(for the definition of this arrow notation and further details, see
2724:
484:
71:
48:
Ceva's theorem, case 2: the three lines are concurrent at a point
30:
Ceva's theorem, case 1: the three lines are concurrent at a point
2684:
2649:
2615:
2734:
2626:-simplex as a ray from each vertex to a point on the opposite (
491:
122:
2841:, pages 177–180, Dover Publishing Co., second revised edition.
2268:, and the right-hand side has the same direction as the line
1767:
But at most one point can cut a segment in a given ratio so
3132:
2818:
Russell, John
Wellesley (1905). "Ch. 1 §7 Ceva's Theorem".
2114:{\displaystyle \lambda _{A}\lambda _{B}\lambda _{C}\neq 0.}
2939:
Samet, Dov (May 2021). "An
Extension of Ceva's Theorem to
3116:
Conics
Associated with a Cevian Nest, by Clark Kimberling
1918:{\displaystyle \lambda _{A}+\lambda _{B}+\lambda _{C}=1,}
468:. The converse is often included as part of the theorem.
260:
is taken to be positive or negative according to whether
2139:(see figures), the last equation may be rearranged into
594:
is outside the triangle (lower diagram shows one case).
517:); cevian nest, anticevian triangle, Ceva conjugate. (
1853:{\displaystyle \lambda _{A},\lambda _{B},\lambda _{C}}
2637:). Then the cevians are concurrent if and only if a
2614:
The theorem can be generalized to higher-dimensional
2468:
2367:
2285:
2148:
2074:
1937:
1869:
1813:
1680:
1636:
The theorem follows by dividing these two equations.
1517:
1378:
1222:
1111:
1000:
772:
606:
337:
138:
268:
in some fixed orientation of the line. For example,
1667:. Then by the theorem, the equation also holds for
2595:
2430:
2347:
2253:
2113:
2042:
1917:
1852:
1756:
1625:
1486:
1340:
1202:
1091:
968:
752:
445:
243:
3181:
2977:
1774:
3092:Derivations and applications of Ceva's Theorem
3012:
2272:. These lines have different directions since
2648:Another generalization to higher-dimensional
305:). It is therefore true for triangles in any
3110:Glossary of Encyclopedia of Triangle Centers
1354:The theorem can also be proven easily using
2852:Hopkins, George Irving (1902). "Art. 986".
3069:
3051:
1639:The converse follows as a corollary. Let
278:is defined as having positive value when
2903:
2683:The analogue of the theorem for general
1213:Multiplying these three equations gives
43:
25:
2851:
2817:
2786:
577:
3182:
3133:
2938:
2868:
2753:
3101:Trigonometric Form of Ceva's Theorem
2813:
2811:
2809:
479:. But it was proven much earlier by
475:, who published it in his 1678 work
471:The theorem is often attributed to
13:
3045:
2609:
1177:
1153:
1066:
1042:
979:(Replace the minus with a plus if
943:
919:
888:
863:
839:
814:
727:
703:
639:
615:
536:, the two theorems may be seen as
14:
3211:
3079:
2945:The American Mathematical Monthly
2906:The American Mathematical Monthly
2871:The American Mathematical Monthly
2806:
2796:Journal for Geometry and Graphics
3015:Journal of Mathematical Sciences
2839:Challenging Problems in Geometry
1647:so that the equation holds. Let
3006:
2837:and Charles T. Salkind (1996),
2758:Geometry: Our Cultural Heritage
2532:
2526:
528:The theorem is very similar to
293:Ceva's theorem is a theorem of
3127:Wolfram Demonstrations Project
2971:
2932:
2897:
2862:
2844:
2828:
2780:
2747:
2720:Menelaus's theorem - Knowledge
2064:For Ceva's theorem, the point
1190:
1173:
1166:
1149:
1079:
1062:
1055:
1038:
956:
939:
932:
915:
901:
884:
876:
859:
852:
835:
827:
810:
740:
723:
716:
699:
652:
635:
628:
611:
483:, an eleventh-century king of
1:
2957:10.1080/00029890.2021.1896292
2740:
2730:Stewart's theorem - Knowledge
1807:are the unique three numbers
1775:Using barycentric coordinates
103:), to meet opposite sides at
2557:
2544:
2493:
2480:
2392:
2379:
1748:
1730:
1705:
1692:
1608:
1595:
1575:
1562:
1542:
1529:
1469:
1456:
1436:
1423:
1403:
1390:
1318:
1305:
1285:
1272:
1252:
1239:
1136:
1123:
1025:
1012:
797:
784:
686:
673:
428:
415:
395:
382:
362:
349:
229:
216:
196:
183:
163:
150:
107:respectively. (The segments
96:(not on one of the sides of
7:
3168:Encyclopedia of Mathematics
2697:
1791:, that belongs to the same
264:is to the left or right of
254:In other words, the length
10:
3216:
513:is the cevian triangle of
127:signed lengths of segments
18:
3156:Dynamic Geometry Sketches
3052:Hogendijk, J. B. (1995).
2459:The same reasoning shows
1497:and from the transversal
987:are on opposite sides of
543:
525:is pronounced chev'ian.)
481:Yusuf Al-Mu'taman ibn Hűd
3200:Euclidean plane geometry
3195:Theorems about triangles
2855:Inductive Plane Geometry
2622:. Define a cevian of an
320:is also true: If points
290:and negative otherwise.
2787:Benitez, Julio (2007).
2620:barycentric coordinates
1797:barycentric coordinates
1358:. From the transversal
582:First, the sign of the
560:barycentric coordinates
521:is pronounced Chay'va;
3071:10.1006/hmat.1995.1001
2858:. D.C. Heath & Co.
2597:
2432:
2349:
2255:
2115:
2044:
1919:
1854:
1758:
1671:. Comparing the two,
1643:be given on the lines
1627:
1488:
1342:
1204:
1093:
970:
754:
558:The second proof uses
447:
328:respectively so that
245:
59:
41:
2835:Alfred S. Posamentier
2754:Holme, Audun (2010).
2715:Circumcevian triangle
2598:
2433:
2350:
2256:
2116:
2045:
1920:
1855:
1759:
1628:
1489:
1343:
1205:
1094:
971:
755:
448:
246:
47:
29:
21:Ceva (disambiguation)
3058:Historia Mathematica
2980:Mathematics Magazine
2762:. Springer. p.
2725:Triangle - Knowledge
2466:
2365:
2283:
2146:
2072:
1935:
1867:
1811:
1678:
1515:
1376:
1220:
1109:
998:
770:
604:
578:Using triangle areas
335:
136:
19:For other uses, see
3125:by Jay Warendorff,
2704:Projective geometry
1779:Given three points
1659:be the point where
498:are the cevians of
316:A slightly adapted
74:. Given a triangle
70:is a theorem about
3135:Weisstein, Eric W.
3027:10.1007/BF01249519
2824:. Clarendon Press.
2735:Cevian - Knowledge
2689:constant curvature
2593:
2428:
2345:
2251:
2111:
2040:
1915:
1850:
1754:
1623:
1484:
1356:Menelaus's theorem
1338:
1200:
1089:
966:
750:
443:
241:
92:to a common point
88:be drawn from the
64:Euclidean geometry
60:
42:
3086:Menelaus and Ceva
2773:978-3-642-14440-0
2709:Median (geometry)
2639:mass distribution
2588:
2561:
2560:
2547:
2530:
2524:
2497:
2496:
2483:
2423:
2396:
2395:
2382:
2358:It follows that
2337:
2309:
2246:
2218:
2190:
2162:
2127:the intersection
2123:If one takes for
2035:
2007:
1979:
1951:
1752:
1751:
1733:
1709:
1708:
1695:
1612:
1611:
1598:
1579:
1578:
1565:
1546:
1545:
1532:
1473:
1472:
1459:
1440:
1439:
1426:
1407:
1406:
1393:
1322:
1321:
1308:
1289:
1288:
1275:
1256:
1255:
1242:
1195:
1140:
1139:
1126:
1084:
1029:
1028:
1015:
961:
906:
801:
800:
787:
745:
690:
689:
676:
657:
530:Menelaus' theorem
432:
431:
418:
399:
398:
385:
366:
365:
352:
233:
232:
219:
200:
199:
186:
167:
166:
153:
3207:
3176:
3148:
3147:
3138:"Ceva's Theorem"
3075:
3073:
3039:
3038:
3021:(4): 3201–3206.
3010:
3004:
3003:
2975:
2969:
2968:
2936:
2930:
2929:
2901:
2895:
2894:
2866:
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2859:
2848:
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2832:
2826:
2825:
2815:
2804:
2803:
2793:
2784:
2778:
2777:
2761:
2751:
2711:– an application
2673:
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2600:
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2126:
2120:
2118:
2117:
2112:
2104:
2103:
2094:
2093:
2084:
2083:
2067:
2056:
2053:for every point
2049:
2047:
2046:
2041:
2036:
2031:
2023:
2021:
2020:
2008:
2003:
1995:
1993:
1992:
1980:
1975:
1967:
1965:
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1952:
1947:
1939:
1924:
1922:
1921:
1916:
1905:
1904:
1892:
1891:
1879:
1878:
1859:
1857:
1856:
1851:
1849:
1848:
1836:
1835:
1823:
1822:
1806:
1803:with respect of
1802:
1790:
1782:
1770:
1763:
1761:
1760:
1755:
1753:
1747:
1743:
1734:
1729:
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1500:
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1380:
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1361:
1347:
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1323:
1317:
1309:
1304:
1296:
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1290:
1284:
1276:
1271:
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1262:
1257:
1251:
1243:
1238:
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1229:
1209:
1207:
1206:
1201:
1196:
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1193:
1176:
1170:
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1152:
1146:
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1135:
1127:
1122:
1114:
1113:
1098:
1096:
1095:
1090:
1085:
1083:
1082:
1065:
1059:
1058:
1041:
1035:
1030:
1024:
1016:
1011:
1003:
1002:
990:
986:
982:
975:
973:
972:
967:
962:
960:
959:
942:
936:
935:
918:
912:
907:
905:
904:
887:
879:
862:
856:
855:
838:
830:
813:
807:
802:
796:
788:
783:
775:
774:
759:
757:
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746:
744:
743:
726:
720:
719:
702:
696:
691:
685:
677:
672:
664:
663:
658:
656:
655:
638:
632:
631:
614:
608:
593:
589:
554:
538:projective duals
516:
512:
501:
497:
477:De lineis rectis
459:
452:
450:
449:
444:
433:
427:
419:
414:
406:
405:
400:
394:
386:
381:
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367:
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353:
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327:
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289:
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95:
87:
80:
58:
51:
40:
33:
3215:
3214:
3210:
3209:
3208:
3206:
3205:
3204:
3190:Affine geometry
3180:
3179:
3161:
3082:
3048:
3046:Further reading
3043:
3042:
3011:
3007:
2992:10.2307/2690569
2976:
2972:
2937:
2933:
2918:10.2307/2300222
2902:
2898:
2883:10.2307/2322390
2877:(10): 936–939.
2867:
2863:
2849:
2845:
2833:
2829:
2816:
2807:
2791:
2785:
2781:
2774:
2752:
2748:
2743:
2700:
2678:Routh's theorem
2668:
2661:
2657:
2653:
2627:
2623:
2612:
2610:Generalizations
2582:
2578:
2572:
2568:
2566:
2549:
2536:
2533:
2527:
2518:
2514:
2508:
2504:
2502:
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2451:
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2445:
2417:
2413:
2407:
2403:
2401:
2384:
2371:
2368:
2366:
2363:
2362:
2326:
2324:
2318:
2314:
2298:
2296:
2290:
2286:
2284:
2281:
2280:
2273:
2269:
2265:
2235:
2233:
2227:
2223:
2207:
2205:
2199:
2195:
2179:
2177:
2171:
2167:
2151:
2149:
2147:
2144:
2143:
2136:
2132:
2128:
2124:
2099:
2095:
2089:
2085:
2079:
2075:
2073:
2070:
2069:
2065:
2054:
2024:
2022:
2016:
2012:
1996:
1994:
1988:
1984:
1968:
1966:
1960:
1956:
1940:
1938:
1936:
1933:
1932:
1900:
1896:
1887:
1883:
1874:
1870:
1868:
1865:
1864:
1844:
1840:
1831:
1827:
1818:
1814:
1812:
1809:
1808:
1804:
1800:
1788:
1780:
1777:
1768:
1736:
1735:
1721:
1717:
1714:
1697:
1684:
1681:
1679:
1676:
1675:
1668:
1664:
1660:
1656:
1652:
1648:
1644:
1640:
1600:
1587:
1584:
1567:
1554:
1551:
1534:
1521:
1518:
1516:
1513:
1512:
1502:
1498:
1461:
1448:
1445:
1428:
1415:
1412:
1395:
1382:
1379:
1377:
1374:
1373:
1363:
1359:
1310:
1297:
1294:
1277:
1264:
1261:
1244:
1231:
1228:
1227:
1223:
1221:
1218:
1217:
1189:
1172:
1171:
1165:
1148:
1147:
1145:
1128:
1115:
1112:
1110:
1107:
1106:
1078:
1061:
1060:
1054:
1037:
1036:
1034:
1017:
1004:
1001:
999:
996:
995:
988:
984:
980:
955:
938:
937:
931:
914:
913:
911:
900:
883:
875:
858:
857:
851:
834:
826:
809:
808:
806:
789:
776:
773:
771:
768:
767:
739:
722:
721:
715:
698:
697:
695:
678:
665:
662:
651:
634:
633:
627:
610:
609:
607:
605:
602:
601:
591:
587:
580:
552:
546:
514:
507:
504:cevian triangle
499:
495:
464:, or all three
457:
420:
407:
404:
387:
374:
371:
354:
341:
338:
336:
333:
332:
325:
321:
295:affine geometry
287:
283:
279:
274:
270:
269:
265:
261:
256:
255:
221:
208:
205:
188:
175:
172:
155:
142:
139:
137:
134:
133:
125:.) Then, using
117:
113:
109:
108:
104:
97:
93:
85:
75:
53:
49:
35:
31:
24:
17:
12:
11:
5:
3213:
3203:
3202:
3197:
3192:
3178:
3177:
3163:"Ceva theorem"
3159:
3149:
3130:
3122:Ceva's Theorem
3118:
3113:
3107:
3098:
3089:
3081:
3080:External links
3078:
3077:
3076:
3047:
3044:
3041:
3040:
3005:
2986:(4): 254–268.
2970:
2951:(5): 435–445.
2931:
2912:(9): 468–472.
2896:
2861:
2843:
2827:
2805:
2779:
2772:
2745:
2744:
2742:
2739:
2738:
2737:
2732:
2727:
2722:
2717:
2712:
2706:
2699:
2696:
2643:center of mass
2611:
2608:
2604:
2603:
2592:
2585:
2581:
2575:
2571:
2565:
2559:
2555:
2552:
2546:
2542:
2539:
2521:
2517:
2511:
2507:
2501:
2495:
2491:
2488:
2482:
2478:
2475:
2439:
2438:
2427:
2420:
2416:
2410:
2406:
2400:
2394:
2390:
2387:
2381:
2377:
2374:
2356:
2355:
2344:
2341:
2336:
2332:
2329:
2321:
2317:
2313:
2308:
2304:
2301:
2293:
2289:
2262:
2261:
2250:
2245:
2241:
2238:
2230:
2226:
2222:
2217:
2213:
2210:
2202:
2198:
2194:
2189:
2185:
2182:
2174:
2170:
2166:
2161:
2157:
2154:
2110:
2107:
2102:
2098:
2092:
2088:
2082:
2078:
2051:
2050:
2039:
2034:
2030:
2027:
2019:
2015:
2011:
2006:
2002:
1999:
1991:
1987:
1983:
1978:
1974:
1971:
1963:
1959:
1955:
1950:
1946:
1943:
1926:
1925:
1914:
1911:
1908:
1903:
1899:
1895:
1890:
1886:
1882:
1877:
1873:
1847:
1843:
1839:
1834:
1830:
1826:
1821:
1817:
1787:, and a point
1776:
1773:
1765:
1764:
1750:
1746:
1742:
1739:
1732:
1727:
1724:
1720:
1713:
1707:
1703:
1700:
1694:
1690:
1687:
1634:
1633:
1622:
1619:
1616:
1610:
1606:
1603:
1597:
1593:
1590:
1583:
1577:
1573:
1570:
1564:
1560:
1557:
1550:
1544:
1540:
1537:
1531:
1527:
1524:
1495:
1494:
1483:
1480:
1477:
1471:
1467:
1464:
1458:
1454:
1451:
1444:
1438:
1434:
1431:
1425:
1421:
1418:
1411:
1405:
1401:
1398:
1392:
1388:
1385:
1349:
1348:
1337:
1334:
1331:
1327:
1320:
1316:
1313:
1307:
1303:
1300:
1293:
1287:
1283:
1280:
1274:
1270:
1267:
1260:
1254:
1250:
1247:
1241:
1237:
1234:
1226:
1211:
1210:
1199:
1192:
1188:
1185:
1182:
1179:
1175:
1168:
1164:
1161:
1158:
1155:
1151:
1144:
1138:
1134:
1131:
1125:
1121:
1118:
1100:
1099:
1088:
1081:
1077:
1074:
1071:
1068:
1064:
1057:
1053:
1050:
1047:
1044:
1040:
1033:
1027:
1023:
1020:
1014:
1010:
1007:
991:.) Similarly,
977:
976:
965:
958:
954:
951:
948:
945:
941:
934:
930:
927:
924:
921:
917:
910:
903:
899:
896:
893:
890:
886:
882:
878:
874:
871:
868:
865:
861:
854:
850:
847:
844:
841:
837:
833:
829:
825:
822:
819:
816:
812:
805:
799:
795:
792:
786:
782:
779:
761:
760:
749:
742:
738:
735:
732:
729:
725:
718:
714:
711:
708:
705:
701:
694:
688:
684:
681:
675:
671:
668:
661:
654:
650:
647:
644:
641:
637:
630:
626:
623:
620:
617:
613:
584:left-hand side
579:
576:
545:
542:
506:(the triangle
454:
453:
442:
439:
436:
430:
426:
423:
417:
413:
410:
403:
397:
393:
390:
384:
380:
377:
370:
364:
360:
357:
351:
347:
344:
324:are chosen on
252:
251:
240:
237:
231:
227:
224:
218:
214:
211:
204:
198:
194:
191:
185:
181:
178:
171:
165:
161:
158:
152:
148:
145:
68:Ceva's theorem
15:
9:
6:
4:
3:
2:
3212:
3201:
3198:
3196:
3193:
3191:
3188:
3187:
3185:
3174:
3170:
3169:
3164:
3160:
3157:
3153:
3150:
3145:
3144:
3139:
3136:
3131:
3128:
3124:
3123:
3119:
3117:
3114:
3111:
3108:
3106:
3102:
3099:
3097:
3093:
3090:
3087:
3084:
3083:
3072:
3067:
3063:
3059:
3055:
3050:
3049:
3036:
3032:
3028:
3024:
3020:
3016:
3009:
3001:
2997:
2993:
2989:
2985:
2981:
2974:
2966:
2962:
2958:
2954:
2950:
2946:
2943:-Simplices".
2942:
2935:
2927:
2923:
2919:
2915:
2911:
2907:
2900:
2892:
2888:
2884:
2880:
2876:
2872:
2865:
2857:
2856:
2847:
2840:
2836:
2831:
2823:
2822:
2821:Pure Geometry
2814:
2812:
2810:
2801:
2797:
2790:
2783:
2775:
2769:
2765:
2760:
2759:
2750:
2746:
2736:
2733:
2731:
2728:
2726:
2723:
2721:
2718:
2716:
2713:
2710:
2707:
2705:
2702:
2701:
2695:
2692:
2690:
2686:
2681:
2679:
2675:
2671:
2664:
2660:-face, in a (
2651:
2646:
2644:
2640:
2636:
2630:
2621:
2617:
2607:
2590:
2583:
2579:
2573:
2569:
2563:
2553:
2550:
2540:
2537:
2519:
2515:
2509:
2505:
2499:
2489:
2486:
2476:
2473:
2462:
2461:
2460:
2457:
2444:
2443:line segments
2425:
2418:
2414:
2408:
2404:
2398:
2388:
2385:
2375:
2372:
2361:
2360:
2359:
2342:
2339:
2334:
2330:
2327:
2319:
2315:
2311:
2306:
2302:
2299:
2291:
2287:
2279:
2278:
2277:
2248:
2243:
2239:
2236:
2228:
2224:
2220:
2215:
2211:
2208:
2200:
2196:
2192:
2187:
2183:
2180:
2172:
2168:
2164:
2159:
2155:
2152:
2142:
2141:
2140:
2131:of the lines
2121:
2108:
2105:
2100:
2096:
2090:
2086:
2080:
2076:
2062:
2060:
2037:
2032:
2028:
2025:
2017:
2013:
2009:
2004:
2000:
1997:
1989:
1985:
1981:
1976:
1972:
1969:
1961:
1957:
1953:
1948:
1944:
1941:
1931:
1930:
1929:
1912:
1909:
1906:
1901:
1897:
1893:
1888:
1884:
1880:
1875:
1871:
1863:
1862:
1861:
1845:
1841:
1837:
1832:
1828:
1824:
1819:
1815:
1798:
1794:
1786:
1783:that are not
1772:
1744:
1740:
1737:
1725:
1722:
1718:
1711:
1701:
1698:
1688:
1685:
1674:
1673:
1672:
1637:
1620:
1617:
1614:
1604:
1601:
1591:
1588:
1581:
1571:
1568:
1558:
1555:
1548:
1538:
1535:
1525:
1522:
1511:
1510:
1509:
1506:
1481:
1478:
1475:
1465:
1462:
1452:
1449:
1442:
1432:
1429:
1419:
1416:
1409:
1399:
1396:
1386:
1383:
1372:
1371:
1370:
1367:
1357:
1352:
1351:as required.
1335:
1332:
1329:
1325:
1314:
1311:
1301:
1298:
1291:
1281:
1278:
1268:
1265:
1258:
1248:
1245:
1235:
1232:
1224:
1216:
1215:
1214:
1197:
1186:
1183:
1180:
1162:
1159:
1156:
1142:
1132:
1129:
1119:
1116:
1105:
1104:
1103:
1086:
1075:
1072:
1069:
1051:
1048:
1045:
1031:
1021:
1018:
1008:
1005:
994:
993:
992:
963:
952:
949:
946:
928:
925:
922:
908:
897:
894:
891:
880:
872:
869:
866:
848:
845:
842:
831:
823:
820:
817:
803:
793:
790:
780:
777:
766:
765:
764:
747:
736:
733:
730:
712:
709:
706:
692:
682:
679:
669:
666:
659:
648:
645:
642:
624:
621:
618:
600:
599:
598:
595:
585:
575:
573:
569:
565:
561:
556:
549:
541:
539:
535:
531:
526:
524:
520:
511:
505:
493:
488:
486:
482:
478:
474:
473:Giovanni Ceva
469:
467:
463:
440:
437:
434:
424:
421:
411:
408:
401:
391:
388:
378:
375:
368:
358:
355:
345:
342:
331:
330:
329:
319:
314:
312:
308:
304:
300:
299:line segments
296:
291:
238:
235:
225:
222:
212:
209:
202:
192:
189:
179:
176:
169:
159:
156:
146:
143:
132:
131:
130:
128:
124:
121:are known as
101:
91:
84:
79:
73:
69:
65:
57:
46:
39:
28:
22:
3166:
3141:
3120:
3105:cut-the-knot
3096:cut-the-knot
3088:at MathPages
3061:
3057:
3018:
3014:
3008:
2983:
2979:
2973:
2948:
2944:
2940:
2934:
2909:
2905:
2899:
2874:
2870:
2864:
2854:
2846:
2838:
2830:
2820:
2799:
2795:
2782:
2757:
2749:
2693:
2682:
2676:
2669:
2662:
2647:
2628:
2613:
2605:
2458:
2440:
2357:
2263:
2122:
2063:
2059:Affine space
2052:
1927:
1778:
1766:
1638:
1635:
1504:
1501:of triangle
1496:
1365:
1362:of triangle
1353:
1350:
1212:
1101:
978:
762:
596:
581:
568:affine plane
557:
550:
547:
534:cross-ratios
527:
522:
518:
509:
503:
489:
476:
470:
455:
315:
307:affine plane
292:
253:
99:
77:
67:
61:
55:
37:
2802:(1): 39–44.
763:Therefore,
494:(the lines
282:is between
3184:Categories
2741:References
1860:such that
1645:BC, AC, AB
496:AD, BE, CF
462:concurrent
458:AD, BE, CF
326:BC, AC, AB
86:AO, BO, CO
81:, let the
3173:EMS Press
3143:MathWorld
3035:123870381
2965:233413469
2650:simplexes
2616:simplexes
2580:λ
2570:λ
2558:¯
2545:¯
2516:λ
2506:λ
2494:¯
2481:¯
2415:λ
2405:λ
2393:¯
2380:¯
2335:→
2316:λ
2307:→
2288:λ
2244:→
2225:λ
2216:→
2197:λ
2188:→
2169:λ
2165:−
2160:→
2106:≠
2097:λ
2087:λ
2077:λ
2033:→
2014:λ
2005:→
1986:λ
1977:→
1958:λ
1949:→
1898:λ
1885:λ
1872:λ
1842:λ
1829:λ
1816:λ
1785:collinear
1749:¯
1731:¯
1706:¯
1693:¯
1618:−
1609:¯
1596:¯
1582:⋅
1576:¯
1563:¯
1549:⋅
1543:¯
1530:¯
1479:−
1470:¯
1457:¯
1443:⋅
1437:¯
1424:¯
1410:⋅
1404:¯
1391:¯
1319:¯
1306:¯
1292:⋅
1286:¯
1273:¯
1259:⋅
1253:¯
1240:¯
1178:△
1154:△
1137:¯
1124:¯
1067:△
1043:△
1026:¯
1013:¯
944:△
920:△
889:△
881:−
864:△
840:△
832:−
815:△
798:¯
785:¯
728:△
704:△
687:¯
674:¯
640:△
616:△
570:over any
429:¯
416:¯
402:⋅
396:¯
383:¯
369:⋅
363:¯
350:¯
309:over any
303:collinear
301:that are
230:¯
217:¯
203:⋅
197:¯
184:¯
170:⋅
164:¯
151:¯
72:triangles
3064:: 1–18.
2850:Follows
2698:See also
2685:polygons
2633:)-face (
1741:′
1726:′
1669:D, E, F'
1663:crosses
1655:and let
1651:meet at
485:Zaragoza
466:parallel
318:converse
90:vertices
52:outside
3175:, 2001
3000:2690569
2926:2300222
2891:2322390
2274:A, B, C
1805:A, B, C
1781:A, B, C
1641:D, E, F
564:vectors
322:D, E, F
123:cevians
105:D, E, F
34:inside
3033:
2998:
2963:
2924:
2889:
2770:
2618:using
1795:, the
1769:F = F’
1649:AD, BE
544:Proofs
523:cevian
492:cevian
3031:S2CID
2996:JSTOR
2961:S2CID
2922:JSTOR
2887:JSTOR
2792:(PDF)
2635:facet
1793:plane
572:field
456:then
311:field
83:lines
2768:ISBN
2450:and
2135:and
1928:and
1102:and
983:and
562:and
519:Ceva
460:are
286:and
3154:at
3103:at
3094:at
3066:doi
3023:doi
2988:doi
2953:doi
2949:128
2914:doi
2879:doi
2764:210
2672:+ 1
2665:+ 1
2631:– 1
2529:and
2061:).
1799:of
1505:BCF
1499:AOD
1366:ACF
1360:BOE
555:.
510:DEF
502:),
100:ABC
78:ABC
62:In
56:ABC
38:ABC
3186::
3171:,
3165:,
3140:.
3062:22
3060:.
3056:.
3029:.
3019:72
3017:.
2994:.
2984:68
2982:.
2959:.
2947:.
2920:.
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2018:C
2010:+
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1998:X
1990:B
1982:+
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1970:X
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392:C
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284:A
280:F
266:Y
262:X
236:=
226:A
223:E
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190:D
180:D
177:B
160:B
157:F
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23:.
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