Centuries passed, and someone proved that the three medians do indeed concur in a point, now called the centroid. The ancients found other points, too, now called the incenter, circumcenter and orthocenter. More centuries passed, more special points were discovered, and a definition of triangle center emerged. Like the definition of continuous function, this definition is satisfied by infinitely many objects, of which only finitely many will ever be published. The Encyclopedia of Triangle Centers (ETC) extends a list of 400 triangle centers published in the book Triangle Centers and Central Triangles. A highly recommended introduction to triangle centers and related geometry is Paul Yiu's A Tour of Triangle Geometry.
Eric Weisstein's MathWorld, the web's most extensive mathematics resource, covers much of classical and modern triangle geometry, including sketches and references. A good place to start is MathWorld's PlaneGeometry. MathWorld is hosted by Wolfram Research, makers of Mathematica. A highly recommended Mathematica package for exploring triangle geometry is described at PlaneGeometry.
NOTATION AND COORDINATES
The reference triangle is ABC, with sidelengths a,b,c. Each triangle center P has homogeneous trilinear coordinates, or simply trilinears, of the form x : y : z. This means there is a nonzero function h of (a,b,c) such that
where x', y', z' denote the directed distances from P to sidelines BC, CA, AB. Likewise, u : v : w are barycentrics if there is a function k of (a,b,c) such that
where u', v', w' denote the signed areas of triangles PBC, PCA, PAB. Both coordinate systems are widely used; if trilinears for a point are x : y : z, then barycentrics are ax : by : cz.
In order that every center should have its own name, in cases where no particular name arises from geometrical or historical considerations, the name of a star is used instead. For example, X(770) is POINT ACAMAR. For a list of star names, visit SkyEye - (Un)Common Star Names.
HOW TO USE ETC
You won't have to scroll down very far to find well known centers. Other named centers can be found using your computer's searcher - for example, search for "Apollonius" to find "Apollonius point" as X(181).
To determine if a possibly new center is already listed, click Search at the top of this page. If you're unsure of a term, click Glossary. For visual constructions of selected centers with text, click Sketches. To learn about the triangle geometry interest group, Hyacinthos, and other resources, or to view acknowledgments or supplementary encyclopedic material, click Links, Thanks, or More.
If you have The Geometer's Sketchpad, you can view sketches of many of the triangle centers. These are dynamic sketches, meaning that you can vary the shape of the reference triangle A, B, C by dragging these vertices. (For information on Sketchpad, click Sketchpad.) The sketches are also useful for making your own Sketchpad tools, so that you can quickly construct X-of-T for many choices of X and T. For example, starting with ABC and point P, you could efficiently construct center X of the four triangles ABC, BCP, CAP, ABP.
X(1) = INCENTER
Trilinears 1 : 1 : 1Barycentrics a : b : c
X(1) is the point of concurrence of the interior angle bisectors of ABC; the point inside ABC whose distances from sidelines BC, CA, AB are equal. This equal distance, r, is the radius of the incircle, given by
Three more points are also equidistant from the sidelines; they are given by these names and trilinears:
The radii of the excircles are
If you have The Geometer's Sketchpad, you can view Incenter.
Writing the A-exradius as r(a,b,c), the others are r(b,c,a) and r(c,a,b). If these exradii are abbreviated as ra, rb, rc, then 1/r = 1/ra + 1/rb + 1/rc. Moreover,
where R denotes the radius of the circumcircle.
The incenter is the identity of the group of triangle centers under "trilinear multiplication" defined by
A construction for * is easily obtained from the construction for "barycentric multiplication" mentioned in connection with X(2), just below.
The incenter and the other classical centers are discussed in these highly recommended books:
Paul Yiu, Introduction to the Geometry of the Triangle, 2002;
Nathan Altshiller Court, College Geometry, Barnes & Noble, 1952;
Roger A. Johnson, Advanced Euclidean Geometry, Dover, 1960.
X(1) lies on these lines:
2,8 3,35 4,33 5,11 6,9 7,20 15,1251 16,1250 19,28 21,31 24,1061 25,1036 29,92 30,79 32,172 39,291 41,101 49,215 60,110 61,203 62,202 64,1439 69,1245 71,579 75,86 76,350 82,560 84,221 87,192 88,100 90,155 99,741 102,108 104,109 142,277 147,150 159,1486 163,293 164,258 166,1488 167,174 168,173 179,1142 181,970 182,983 184,1726 185,296 188,361 190,537 196,207 201,212 204,1712 224,377 227,1465 228,1730 229,267 256,511 257,385 280,1256 281,282 289,363 312,1089 318,1897 320,752 321,964 329,452 335,384 336,811 341,1050 344,1265 346,1219 357,1508 358,1507 364,365 371,1702 372,1703 376,553 378,1063 393,836 394,1711 409,1247 410,1248 411,1254 442,1834 474,1339 475,1861 512,875 513,764 514,663 522,1459 528,1086 561,718 563,1820 564,1048 572,604 573,941 574,1571 594,1224 607,949 631,1000 644,1280 647,1021 650,1643 651,1156 659,891 662,897 672,1002 689,719 704,1502 727,932 731,789 748,756 761,825 765,1052 810,1577 840,1308 905,1734 908,998 921,1800 939,1260 945,1875 947,1753 951,1435 969,1444 971,1419 989,1397 1013,1430 1037,1041 1053,1110 1057,1598 1059,1597 1075,1148 1106,1476 1168,1318 1170,1253 1185,1206 1197,1613 1292,1477 1333,1761 1342,1700 1343,1701 1361,1364 1389,1393 1399,1727 1406,1480 1409,1765 1437,1710 1472,1791 1719,1790 1855,1886 1859,1871 1872,1887
X(1) is the {X(2),X(8)}-harmonic conjugate of X(10). For a list of other harmonic conjugates of X(1), click More at the top of this page.
X(1) = midpoint of X(I) and X(J) for these (I,J): (7,390), (8,145)
X(1) = reflection of X(I) in X(J) for these (I,J): (2,551), (3,1385), (4,946), (6,1386), (8,10), (9,1001), (10,1125), (11,1387), (36,1319), (40,3), (43,995), (46,56), (57,999), (63,993), (65,942), (72,960), (80,11), (100,214), (191,21), (200,997), (238,1279), (267,229), (291,1015), (355,5), (484,36), (984,37), (1046,58), (1054,106), (1478,226)
X(1) = isogonal conjugate of X(1)
X(1) = isotomic conjugate of X(75)
X(1) = cyclocevian conjugate of X(1029)
X(1) = inverse-in-circumcircle of X(36)
X(1) = inverse-in-Fuhrmann-circle of X(80)
X(1) = inverse-in-Bevan-circle of X(484)
X(1) = complement of X(8)
X(1) = anticomplement of X(10)
X(1) = anticomplementary conjugate of X(1330)
X(1) = complementary conjugate at X(1329)
X(1) = eigencenter of cevian triangle of X(I) for I = 1, 88, 162
X(1) = eigencenter of anticevian triangle of X(I) for I = 1, 44, 513
X(1) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,9), (4,46), (6,43), (7,57), (8,40), (9,165), (10,191), (21,3), (29,4), (75,63), (77,223), (78,1490), (80,484), (81,6), (82,31), (85,169), (86,2), (88,44), (92,19), (100,513), (104,36), (105,238), (174,173), (188,164), (220,170), (259,503), (266,361), (280,84), (366,364), (508,362), (1492,1491)
X(1) = cevapoint of X(I) and X(J) for these (I,J):
(2,192), (6,55), (11,523), (15,202), (16,203), (19,204), (31,48), (34,207), (37,42), (50,215), (56,221), (65,73), (78,1490), (244,513)
X(1) = X(I)-cross conjugate of X(J) for these (I,J):
(2,87), (3,90), (6,57), (31,19), (33,282), (37,2), (38,75), (42,6), (44,88), (48,63), (55,9), (56,84), (58,267), (65,4), (73,3), (192,43), (207,1490), (221,40), (244,513), (259,258), (266,505), (354,7), (367,366), (500,35), (513,100), (517,80), (518,291), (1491,1492)
X(1) = crosspoint of X(I) and X(J) for these (I,J):
(2,7), (8,280), (21,29), (59,110), (75,92), (81,86)
X(1) = crosssum of X(I) and X(J) for these (I,J):
(2,192), (4,1148), (6,55), (11,523), (15,202), (16,203), (19,204), (31,48), (34,207), (37,42), (44,678), (50,215), (56,221), (57,1419), (65,73), (207,1490), (214,758), (244,513), (500,942), (512,1015), (774,820), (999,1480)
X(1) = crossdifference of any two points on line X(44)X(513)
X(1) = X(I)-Hirst inverse of X(J) for these (I,J):
(2,239), (4,243), (6,238), (9,518), (19,240), (57,241), (105,294), (291,292).
X(1) = X(6)-line conjugate of X(44)
X(1) = X(I)-aleph conjugate of X(J) for these (I,J):
(1,1), (2,63), (4,920), (21,411), (29,412), (88,88), (100,100), (162,162), (174,57), (188,40), (190,190), (266,978), (365,43), (366,9), (507,173), (508,169), (513,1052), (651, 651), (653,653), (655,655), (658,658), (660,660), (662,662), (673,673), (771,771), (799,799), (823,823), (897,897)
X(1) = X(I)-beth conjugate of X(J) for these (I,J):
(1,56), (2,948), (8,8), (9,45), (21,1), (29,34), (55,869), (99,85), (100,1), (110,603), (162,208), (643,1), (644,1), (663,875), (664,1), (1043,78)
X(2) = CENTROID
Trilinears 1/a : 1/b : 1/c= bc : ca : ab
= csc A : csc B : csc C
= cos A + cos B cos C : cos B + cos C cos A : cos C + cos A cos B
= sec A + sec B sec C : sec B + sec C sec A : sec C + sec A sec B
= cos A + cos(B - C) : cos B + cos(C - A) : cos C + cos(A - B)
= cos B cos C - cos(B - C) : cos C cos A - cos(C - A) : cos A cos B - cos(A - B)
Barycentrics 1 : 1 : 1
X(2) is the point of concurrence of the medians of ABC, situated 1/3 of the distance from each vertex to the midpoint of the opposite side. More generally, if L is any line in the plane of ABC, then the distance from X(2) to L is the average of the distances from A, B, C to L. An idealized triangular sheet balances atop a pin head located at X(2) and also balances atop any knife-edge that passes through X(2). The triangles BXC, CXA, AXB have equal areas if and only if X = X(2).
If you have The Geometer's Sketchpad, you can view Centroid.
X(2) is the centroid of the set of 3 vertices, the centroid of the triangle including its interior, but not the centroid of the triangle without its interior; that centroid is X(10).
X(2) is the identity of the group of triangle centers under "barycentric multiplication" defined by
X(2) lies on these lines:
1,8 3,4 6,69 7,9 11,55 12,56 13,16 14,15 17,62 18,61 19,534 31,171 32,83 33,1040 34,1038 35,1479 36,535 37,75 38,244 39,76 40,946 44,89 45,88 51,262 52,1216 54,68 58,540 65,959 66,206 71,1246 72,942 74,113 77,189 80,214 85,241 92,273 94,300 95,97 98,110 99,111 101,116 102,117 103,118 104,119 106,121 107,122 108,123 109,124 112,127 128,1141 129,1298 130,1303 131,1300 133,1294 136,925 137,930 154,1503 165,516 169,1763 174,236 176,1659 178,188 187,316 196,653 201,1393 210,354 216,232 220,1170 222,651 231,1273 242,1851 243,1857 252,1166 253,1073 254,847 257,1432 261,593 265,1511 271,1034 272,284 280,318 283,580 290,327 292,334 294,949 308,702 311,570 314,941 319,1100 322,1108 330,1107 341,1219 351,804 355,944 360,1115 366,367 371,486 372,485 392,517 476,842 480,1223 489,1132 490,1131 495,956 496,1058 514,1022 523,1649 525,1640 561,716 568,1154 572,1746 573,1730 578,1092 585,1336 586,1123 588,1504 589,1505 594,1255 647,850 648,1494 650,693 664,1121 668,1015 670,1084 689,733 743,789 799,873 812,1635 846,1054 914,1442 918,1638 927,1566 954,1260 968,1738 1000,1145 1043,1834 1060,1870 1074,1785 1076,1838 1089,1224 1093,1217 1124,1378 1143,1489 1155,1836 1171,1509 1186,1207 1257,1265 1284,1403 1335,1377 1340,1349 1341,1348 1500,1574 1501,1691 1672,1681 1673,1680 1674,1679 1675,1678 1697,1706
X(2) is the {X(3),X(5)}-harmonic conjugate of X(4). For a list of other harmonic conjugates of X(2), click More at the top of this page.
X(2) = midpoint of X(I) and X(J) for these (I,J): (3,381), (4,376), (210,354)
X(2) = reflection of X(I) in X(J) for these (I,J): (1,551), (3,549), (4,381), (5,547), (6,597), (20,376), (69,599), (148,671), (376,3), (381,5), (549,140), (551,1125), (599,141), (671,115), (903,1086), (1121,1146)
X(2) = isogonal conjugate of X(6)
X(2) = isotomic conjugate of X(2)
X(2) = cyclocevian conjugate of X(4)
X(2) = inverse-in-circumcircle of X(23)
X(2) = inverse-in-nine-point-circle of X(858)
X(2) = inverse-in-Brocard-circle of X(110)
X(2) = complement of X(2)
X(2) = anticomplement of X(2)
X(2) = anticomplementary conjugate of X(69)
X(2) = complementary conjugate of X(141)
X(2) = X(I)-Ceva conjugate of X(J) for these (I,J):
(1,192), (4,193), (6,194), (7,145), (8,144), (30,1494), (69,20), (75,8), (76,69), (83,6), (85,7), (86,1), (87,330), (95,3), (98,385), (99,523), (190,514), (264,4), (274,75), (276, 264), (287,401), (290,511), (308,76), (312,329), (325,147), (333,63), (348,347), (491,487), (492,488), (523,148), (626,1502)
X(2) = cevapoint of X(I) and X(J) for these (I,J):
(1,9), (3,6), (5,216), (10,37), (32,206), (39,141), (44,214), (57,223), (114,230), (115,523), (128,231), (132,232), (140,233), (188,236)
X(2) = X(I)-cross conjugate of X(J) for these (I,J):
(1,7), (3,69), (4,253), (5,264), (6,4), (9,8), (10,75), (32,66), (37,1), (39,6), (44,80), (57,189), (75,330), (114,325), (140,95), (141,76), (142,85), (178,508), (187,67), (206,315), (214,320), (216,3), (223,329), (226,92), (230,98), (233,5), (281,280), (395,14), (396,13), (440,306), (511,290), (514,190), (523,99)
X(2) = crosspoint of X(I) and X(J) for these (I,J):
(1,87), (75,85), (76,264), (83,308), (86,274), (95,276)
X(2) = crosssum of X(I) and X(J) for these (I,J):
(1,43), (2,194), (31,41), (32,184), (42,213), (51,217), (125,826), (649,1015), (688,1084), (902,1017), (1400,1409)
X(2) = crossdifference of any two points on line X(187)X(237)
X(2) = X(I)-Hirst inverse of X(J) for these (I,J):
(1,239), (3,401), (4,297), (6,385), (21,448), (27,447), (69,325), (75,350), (98,287), (115,148), (193,230), (291,335), (298,299), (449,452)
X(2) = X(3)-line conjugate of X(237) = X(316)-line conjugate of X(187)
X(2) = X(I)-aleph conjugate of X(J) for these (I,J):
(1,1045), (2,191), (86,2), (174,1046), (333,20), (366,846)
X(2) = X(I)-beth conjugate of X(J) for these (I,J):
(2,57), (21,995) (190,2), (312,312), (333,2), (643,55), (645,2), (646,2), (648,196), (662,222)
X(2) is the unique point X (as a function of a,b,c) for which the vector-sum XA + XB + XC is the zero vector.
X(3) = CIRCUMCENTER
Trilinears cos A : cos B : cos C= a(b2 + c2 - a2) : b(c2 + a2 - b2) : c(a2 + b2 - c2)
Barycentrics sin 2A : sin 2B : sin 2C
X(3) is the point of concurrence of the perpendicular bisectors of the sides of ABC. The lengths of segments AX, BX, CX are equal if and only if X = X(3). This common distance is the radius of the circumcircle, which passes through vertices A,B,C. Called the circumradius, it is given by
If you have The Geometer's Sketchpad, you can view Circumcenter.
X(3) lies on these lines:
1,35 2,4 6,15 7,943 8,100 9,84 10,197 11,499 12,498 13,17 14,18 19,1871 31,601 33,1753 34,1465 37,975 38,976 41,218 42,967 47,1399 48,71 49,155 54,97 60,1175 63,72 64,154 66,141 67,542 68,343 69,332 73,212 74,110 76,98 77,1410 83,262 86,1246 90,1898 95,264 101,103 102,109 105,277 106,1293 107,1294 108,1295 111,1296 112,1297 113,122 114,127 119,123 125,131 128,1601 142,516 143,1173 145,1483 149,1484 158,243 161,1209 169,910 191,1768 193,1353 194,385 200,963 201,1807 207,1767 223,1035 225,1074 227,1455 238,978 252,930 256,987 269,939 296,820 298,617 299,616 302,621 303,622 305,1799 315,325 345,1791 347,1119 348,1565 352,353 388,495 390,1058 393,1217 395,398 396,397 476,477 485,590 486,615 489,492 490,491 496,497 525,878 595,995 611,1469 613,1428 618,635 619,636 623,629 624,630 639,641 640,642 653,1148 662,1098 667,1083 691,842 695,1613 847,925 901,953 902,1201 917,1305 920,1858 934,972 945,1457 950,1210 951,1407 955,1170 960,997 962,1621 1000,1476 1033,1249 1037,1066 1054,1283 1055,1334 1057,1450 1093,1105 1167,1413 1177,1576 1180,1627 1184,1194 1196,1611 1298,1303 1331,1797 1364,1795 1397,1682 1398,1870 1406,1464 1411,1772 1427,1448 1452,1905 1728,1864 1737,1837 1770,1836 1779,1780
X(3) is the {X(2),X(4)}-harmonic conjugate of X(5). For a list of other harmonic conjugates of X(3), click More at the top of this page. If triangle ABC is acute, then X(3) is the incenter of the tangential triangle and the Bevan point, X(40), of the orthic triangle.
X(3) = midpoint of X(I) and X(J) for these (I,J):
(1,40), (2,376), (4,20), (22,378), (74,110), (98,99), (100,104), (101,103), (102,109), (476,477)
X(3) = reflection of X(I) in X(J) for these (I,J):
(1,1385), (2,549), (4,5), (5,140), (6,182), (20,550), (52,389), (110,1511), (114,620), (145,1483), (149,1484), (155,1147), (193,1353), (195,54), (265,125), (355,10), (381,2), (382,4), (399,110), (550,548), (576,575), (946,1125), (1351,6), (1352,141), (1482,1)
X(3) = isogonal conjugate of X(4)
X(3) = isotomic conjugate of X(264)
X(3) = inverse-in-nine-point-circle of X(2072)
X(3) = inverse-in-orthocentroidal-circle of X(5)
X(3) = inverse-in-1st-Lemoine-circle of X(2456)
X(3) = inverse-in-2nd-Lemoine-circle of X(1570)
X(3) = complement of X(4)
X(3) = anticomplement of X(5)
X(3) = complementary conjugate of X(5)
X(3) = eigencenter of the medial triangle
X(3) = eigencenter of the tangential triangle
X(3) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,6), (4,155), (5,195), (20,1498), (21,1), (22,159), (30,399), (63,219), (69,394), (77,222), (95,2), (96,68), (99,525), (100,521), (110,520), (250, 110), (283,255)
X(3) = cevapoint of X(I) and X(J) for these (I,J):
(6,154), (48,212), (55,198), (71,228), (185,417), (216,418)
X(3) = X(I)-cross conjugate of X(J) for these (I,J):
(48,222), (55,268), (71,63), (73,1), (184,6), (185,4), (212,219), (216,2), (228,48), (520,110)
X(3) = crosspoint of X(I) and X(J) for these (I,J):
(1,90), (2,69), (4,254), (21,283), (54,96), (59,100), (63,77), (78,271), (81,272), (95,97), (99,249), (110,250), (485,486)
X(3) = crosssum of X(I) and X(J) for these (I,J):
(1,46), (2,193), (3,155), (5,52), (6,25), (11,513), (19,33), (30,113), (37,209), (39, 211), (51,53), (65,225), (114,511), (115,512), (116,514), (117, 515), (118,516), (119,517), (120,518), (121,519), (122,520), (123,521), (124,522), (125,523), (126,524), (127,525), (128,1154), (136,924), (184,571), (185,235), (371,372), (487,488)
X(3) = crossdifference of any two points on line X(230)X(231)
X(3) = X(I)-Hirst inverse of X(J) for these (I,J): (2, 401), (4,450), (6,511), (21,416), (194, 385)
X(3) = X(2)-line conjugate of X(468)
X(3) = X(I)-aleph conjugate of X(J) for these (I,J): (1,1046), (21,3), (188,191), (259,1045)
X(3) = X(I)-beth conjugate of X(J) for these (I,J):
(3,603), (8,355), (21,56), (78,78), (100,3), (110,221), (271,84), (283,3), (333,379), (643,8)
X(4) = ORTHOCENTER
Trilinears sec A : sec B : sec C= cos A - sin B sin C : cos B - sin C sin A : cos C - sin A sinB
= cos A - cos(B - C) : cos B - cos(C - A) : cos C - cos(A - B)
= sin B sin C - cos(B - C) : sin C sin A - cos(C - A) : sin A sin B - cos(A - B)
Barycentrics tan A : tan B : tan C
X(4) is the point of concurrence of the altitudes of ABC.
If you have The Geometer's Sketchpad, you can view Orthocenter.
X(4) and the vertices A,B,C comprise an orthocentric system, defined as a set of four points, one of which is the orthocenter of the triangle of the other three (so that each is the orthocenter of the other three). The incenter and excenters also form an orthocentric system. If ABC is acute, then X(4) is the incenter of the orthic triangle.
Suppose P is not on a sideline of ABC. Let A'B'C' be the cevian triangle of P. Let D,E,F be the circles having diameters AA', BB',CC'. The radical center of D,E,F is X(4). (Floor van Lamoen, Hyacinthos #214, Jan. 24, 2000.)
Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 2: The Orthocenter.
X(4) lies on these lines:
1,33 2,3 6,53 7,273 8,72 9,10 11,56 12,55 13,61 14,62 15,17 16,18 32,98 35,498 36,499 37,1841 39,232 42,1860 46,90 48,1881 49,156 51,185 52,68 54,184 57,84 64,1853 65,158 67,338 69,76 74,107 78,908 79,1784 80,1825 83,182 93,562 94,143 96,231 99,114 100,119 101,118 102,124 103,116 109,117 110,113 111,1560 120,1292 121,1293 122,1294 123,1295 126,1296 127,1289 128,930 129,1303 130,1298 131,135 137,933 141,1350 145,149 147,148 150,152 155,254 162,270 165,1698 171,601 193,1351 195,399 204,1453 218,294 238,602 240,256 250,1553 252,1487 276,327 279,1565 371,485 372,486 390,495 394,1217 477,1304 487,489 488,490 496,999 512,879 523,1552 542,576 569,1179 572,1474 574,1506 575,598 579,1713 580,1714 590,1151 608,1518 615,1152 616,627 617,628 653,1156 774,1254 801,1092 842,935 937,1534 940,1396 941,1880 953,1309 1036,1065 1037,1067 1038,1076 1039,1096 1040,1074 1160,1162 1161,1163 1251,1832 1329,1376 1340,1348 1341,1349 1385,1538 1430,1468 1499,1550 1715,1730 1716,1721 1717,1718 1726,1782
X(4) is the {X(3),X(5)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(4), click More at the top of this page.
X(4) = midpoint of X(I) and X(J) for these (I,J):
(3,382), (147,148), (149,153), (150,152)
X(4) = reflection of X(I) in X(J) for these (I,J): (1,946), (2,381), (3,5), (5,546), (8,355), (20,3), (24,235), (25,1596), (40,10), (69,1352), (74,125), (98,115), (99,114), (100,119), (101,118), (102,124), (103,116), (104,11), (107,133), (109,117), (110,113), (112,132), (145,1482), (185,389), (186,403), (193,1351), (376,2), (378,427), (550,140), (925,131), (930,128), (944,1), (1113,1312), (1114,1313), (1141,137), (1292,120), (1293,121), (1294,122), (1295,123), (1296,126), (1297,127), (1298,130), (1299,135), (1300,136), (1303, 129), (1350,141), (1593,1595)
X(4) = isogonal conjugate of X(3)
X(4) = isotomic conjugate of X(69)
X(4) = cyclocevian conjugate of X(2)
X(4) = inverse-in-circumcircle of X(186)
X(4) = inverse-in-nine-point-circle of X(403)
X(4) = complement of X(20)
X(4) = anticomplement of X(3)
X(4) = anticomplementary conjugate of X(20)
X(4) = eigencenter of cevian triangle of X(I) for I = 1, 88, 162
X(4) = eigencenter of anticevian triangle of X(I) for I = 1, 44, 513
X(4) = X(I)-Ceva conjugate of X(J) for these (I,J):
(7,196), (27,19), (29,1), (92,281), (107,523), (264,2), (273,278), (275,6), (286,92), (393, 459)
X(4) = cevapoint of X(I) and X(J) for these (I,J):
(1,46), (2,193), (3,155), (5,52), (6,25), (11,513), (19,33), (30,113), (34,208), (37,209), (39,211), (51,53), (65,225), (114,511), (115,512), (116,514), (117,515), (118,516), (119,517), (120,518), (121,519), (122,520), (123,521), (124,522), (125,523), (126,524), (127,525), (185,235), (371,372), (487,488)
X(4) = X(I)-cross conjugate of X(J) for these (I,J):
(3,254), (6,2), (19,278), (25,393), (33,281), (51,6), (52,24), (65,1), (113,403), (125,523), (185,3), (193,459), (225,158), (389,54), (397,17), (398,18), (407,225), (427,264), (512,112), (513,108), (523,107)
X(4) = crosspoint of X(I) and X(J) for these (I,J): (2,253), (7,189), (27,286), (92,273)
X(4) = crosssum of X(I) and X(J) for these (I,J):
(4,1075), (6,154), (25,1033), (48,212), (55,198), (56,1035), (71,228), (184,577), (185,417), (216,418)
X(4) = crossdifference of any two points on line X(520)X(647)
X(4) = X(I)-Hirst inverse of X(J) for these (I,J):
(1,243), (2,297), (3,350), (19,242), (21,425), (24,421), (25,419), (27,423), (28,422), (29,415), (420,427), (424,451), (459,460), (470,471), (1249,1503)
X(4) = X(I)-aleph conjugate of X(J) for these (I,J): (1,1047), (29,4)
X(4) = X(I)-beth conjugate of X(J) for these (I,J):
(4,34), (8,40), (29,4), (162,56), (318,318), (811,331)
X(5) = NINE-POINT CENTER
Trilinears cos(B - C) : cos(C - A) : cos(A - B)= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A + 2 cos B cos C
= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = cos A - 2 sin B sin C
= h(a,b,c) : h(b,c,a): h(c,a,b), where h(a,b,c) = bc[a2(b2 + c2) - (b2 - c2)2]
Barycentrics a cos(B - C) : b cos(C - A) : c cos(A - B)
= h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = a2(b2 + c2) - (b2 - c2)2
X(5) is the center of the nine-point circle. Euler showed in that this circle passes through the midpoints of the sides of ABC and the feet of the altitudes of ABC, hence six of the nine points. The other three are the midpoints of segments A-to-X(4), B-to-X(4), C-to-X(4). The radius of the nine-point circle is one-half the circumradius.
Dan Pedoe, Circles: A Mathematical View, Mathematical Association of America, 1995.
If you have The Geometer's Sketchpad, you can view Nine-point center and Euler Line.
X(5) lies on these lines:
1,11 2,3 6,68 8,1389 9,1729 10,517 13,18 14,17 15,2913 16,2912 32,230 33,1062 34,1060 39,114 40,1698 46,1836 49,54 51,52 53,216 55,498 56,499 57,1728 65,1737 69,1351 72,908 76,262 79,1749 83,98 85,1565 96,1166 113,125 116,118 117,124 122,133 127,132 128,137 129,130 131,136 141,211 142,971 156,184 182,206 183,315 217,1625 225,1465 226,912 252,1157 264,1093 298,634 299,633 302,622 303,621 311,1225 316,1078 339,1235 371,590 372,615 386,1834 388,999 392,1512 491,637 492,638 515,1125 524,576 539,1493 542,575 570,1879 573,1213 578,1147 579,1901 582,1754 601,750 602,748 618,629 619,630 842,1287 920,1454 1073,1217 1090,1091 1155,1770 1173,1487 1181,1899 1214,1838 1498,1853 1848,1871 1861,1872
X(5) is the {X(2),X(4)}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(5), click More at the top of this page.
X(5) = midpoint of X(I) and X(J) for these (I,J):
(1,355), (2,381), (3,4), (11,119), (20,382), (68,155), (110,265), (113,125), (114,115), (116,118), (117,124), (122,133), (127,132), (128,137), (129,130), (131,136)
X(5) = reflection of X(I) in X(J) for these (I,J): (2,547), (3,140), (4,546), (20,548), (52,143), (549,2), (550,3), (1263,137), (1353,6), (1385,1125), (1483,1), (1484,11)
X(5) = isogonal conjugate of X(54)
X(5) = isotomic conjugate of X(95)
X(5) = inverse-in-circumcircle of X(2070)
X(5) = inverse-in-orthocentroidal-circle of X(3)
X(5) = complement of X(3)
X(5) = anticomplement of X(140)
X(5) = complementary conjugate of X(3)
X(5) = eigencenter of anticevian triangle of X(523)
X(5) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,216), (4,52), (110,523), (264, 324), (265,30), (311,343), (324,53)
X(5) = cevapoint of X(I) and X(J) for these (I,J): (3,195), (51,216)
X(5) = X(I)-cross conjugate of X(J) for these (I,J): (51,53), (216,343), (233,2)
X(5) = crosspoint of X(I) and X(J) for these (I,J): (2,264), (311,324)
X(5) = crosssum of X(I) and X(J) for these (I,J): (3,1147), (6,184)
X(5) = crossdifference of any two points on line X(50)X(647)
X(5) = X(1)-aleph conjugate of X(1048)
X(6) = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT)
Trilinears a : b : c= sin A : sin B : sin C
Barycentrics a2 : b2 : c2
X(6) is the point of concurrence of the symmedians (reflections of medians in corresponding angle bisectors); the point (x, y, z), given here in actual trilinear distances, that minimizes x2 + y2 + z2.
If you have The Geometer's Sketchpad, you can view Symmedian point.
Let A'B'C' be the pedal triangle of an arbitrary point X, and let S(X) be the vector sum XA' + XB' + XC'. Then
X(6) is the radical trace of the 1st and 2nd Lemoine circles. (Peter J. C. Moses, 8/24/03)
Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 7: The Symmedian Point.
X(6) lies on these lines:
1,9 2,69 3,15 4,53 5,68 7,294 8,594 10,1377 13,14 17,18 19,34 21,941 22,251 23,353 24,54 25,51 26,143 27,1246 31,42 33,204 36,609 40,380 41,48 43,87 57,222 60,1169 64,185 66,427 67,125 70,1594 74,112 75,239 76,83 77,241 88,89 98,262 99,729 100,739 101,106 105,1002 110,111 145,346 157,248 160,237 162,1013 169,942 181,197 190,192 194,384 210,612 226,1751 256,1580 264,287 274,1218 279,1170 281,1146 282,1256 291,985 292,869 297,317 305,1241 314,981 330,1258 344,1332 354,374 442,1714 493,1583 494,1584 513,1024 517,998 519,996 523,879 560,1631 561,720 588,1599 589,1600 593,1171 595,1126 598,671 603,1035 644,1120 657,1459 662,757 688,882 689,703 691,843 692,1438 694,1084 706,1502 717,789 750,899 753,825 755,827 840,919 846,1051 893,1403 909,1415 911,1461 939,1802 943,1612 947,1622 959,961 963,1208 967,1790 971,990 986,1046 1096,1859 1112,1177 1131,1132 1139,1140 1166,1601 1173,1614 1174,1617 1195,1399 1201,1696 1214,1708 1327,1328 1362,1416 1399,1425 1423,1429 1718,1781 1826,1837 1836,1839 1854,1858
X(6) is the {X(15),X(16)}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(6), click More at the top of this page.
X(6) = midpoint of X(69) and X(193)
X(6) = reflection of X(I) in X(J) for these (I,J): (1,1386), (2,597), (3,182), (67,125), (69,141), (159,206), (182,575), (592,2), (694,1084), (1350,3), (1351,576), (1352,5)
X(6) = isogonal conjugate of X(2)
X(6) = isotomic conjugate of X(76)
X(6) = cyclocevian conjugate of X(1031)
X(6) = inverse-in-circumcircle of X(187)
X(6) = inverse-in-orthocentroidal-circle of X(115)
X(6) = inverse-in-1st-Lemoine-circle of X(1691)
X(6) = complement of X(69)
X(6) = anticomplement of X(141)
X(6) = anticomplementary conjugate of X(1369)
X(6) = complementary conjugate of X(1368)
X(6) = X(I)-Ceva conjugate of X(J) for these (I,J):
(1,55), (2,3), (3,154), (4,25), (7,1486), (8,197), (9,198), (10,199), (54,184), (57, 56), (58,31), (68,161), (69,159), (74,1495), (76,22), (81,1), (83,2), (88,36), (95,160), (98,237), (110,512), (111,187), (222,221), (249,110), (251,32), (264,157), (275,4), (284,48), (287,1503), (288,54), (323,399), (394,1498)
X(6) = cevapoint of X(I) and X(J) for these (I,J): (1,43), (2,194), (31,41), (32,184), (42,213), (51,217)
X(6) = X(I)-cross conjugate of X(J) for these (I,J):
(25,64), (31,56), (32,25), (39,2), (41,55), (42,1), (51,4), (184,3), (187,111), (213,31), (217,184), (237,98), (512,110)
X(6) = crosspoint of X(I) and X(J) for these (I,J):
(1,57), (2,4), (9,282), (54,275), (58,81), (83, 251), (110,249), (266,289)
X(6) = crosssum of X(I) and X(J) for these (I,J):
(1,9), (3,6), (4,1249), (5,216), (10,37), (11,650), (32,206), (39,141), (44,214), (56,478), (57,223), (114,230), (115,523), (125,647), (128,231), (132,232), (140,233), (142,1212), (188,236), (226,1214), (244,661), (395,619), (396, 618), (512,1084), (513,1015), (514,1086), (522,1146), (570,1209), (577,1147), (590,641), (615,642), (1125,1213), (1196,1368)
X(6) = crossdifference of any two points on line X(30)X(511)
X(6) = X(I)-Hirst inverse of X(J) for these (I,J): (1,238), (2,385), (3,511), (15,16), (25,232), (56,1458), (58,1326), (523,1316), (1423,1429)
X(6) = X(I)-line conjugate of X(J) for these (I,J): (1,518), (2,524), (3,511)
X(6) = X(I)-aleph conjugate of X(J) for these (I,J): (1,846), (81,6), (365,1045), (366,191), (509,1046)
X(6) = X(I)-beth conjugate of X(J) for these (I,J):
(6,604), (9,9), (21,1001), (101,6), (284,6), (294,6), (644,6), (645,6), (651,6), (652,7), (666,6)
X(7) = GERGONNE POINT
Trilinears bc/(b + c - a) : ca/(c + a - b) : ab/(a + b - c)= sec2(A/2) : sec2(B/2) : sec2(C/2)
Barycentrics 1/(b + c - a) : 1/(c + a - b) : 1/(a + b - c)
Let A', B', C' denote the points in which the incircle meets the sides BC, CA, AB, respectively. The lines AA', BB', CC' concur in X(7).
If you have The Geometer's Sketchpad, you can view Gergonne point.
X(7) lies on these lines:
1,20 2,9 3,943 4,273 6,294 8,65 11,658 12,1268 21,56 27,81 37,241 33,1041 34,1039 58,272 59,1275 72,443 73,1246 76,1479 80,150 92,189 100,1004 104,934 108,1013 109,675 145,1266 171,983 174,234 177,555 190,344 192,335 193,239 218,277 220,1223 225,969 238,1471 253,280 256,982 274,959 281,653 286,331 310,314 330,1432 349,1269 354,479 404,1259 452,1467 464,1214 480,1376 492,1267 513,885 517,1000 528,664 554,1082 594,599 604,1429 757,1414 840,927 857,1901 870,1431 940,1407 941,1427 944,1389 952,1159 986,1254 987,1106 1002,1362 1020,1765 1061,1870 1354,1367 1365,1366 1386,1456 1419,1449 1435,1848 1486,1602 1617,1621
X(7) is the {X(69),X(75)}-harmonic conjugate of X(8). For a list of other harmonic conjugates of X(7), click More at the top of this page.
X(7) = reflection of X(I) in X(J) for these (I,J): (9,142), (144,9), (390,1), (673,1086), (1156,11)
X(7) = isogonal conjugate of X(55)
X(7) = isotomic conjugate of X(8)
X(7) = cyclocevian conjugate of X(7)
X(7) = inverse-in-incircle of (1323)
X(7) = complement of X(144)
X(7) = anticomplement of X(9)
X(7) = anticomplementary conjugate of X(329)
X(7) = X(I)-Ceva conjugate of X(J) for these (I,J): (75,347), (85,2), (86,77), (286,273), (331,278)
X(7) = cevapoint of X(I) and X(J) for these (I,J):
(1,57), (2,145), (4,196), (11,514), (55,218), (56,222), (63,224), (65,226), (81,229), (177,234)
X(7) = X(I)-cross conjugate of X(J) for these (I,J):
(1,2), (4,189), (11,514), (55,277), (56,278), (57,279), (65,57), (177,174), (222,348), (226,85), (354,1), (497,8), (517,88)
X(7) = crosspoint of X(I) and X(J) for these (I,J): (75,309), (86,286)
X(7) = crosssum of X(I) and X(J) for these (I,J): (41,1253), (42,228)
X(7) = crossdifference of any two points on line X(657)X(663)
X(7) = X(57)-Hirst inverse of X(1447)
X(7) = X(I)-beth conjugate of X(J) for these (I,J):
(7,269), (21,991), (75,75), (86,7), (99,7), (190,7), (290,7), (314,69), (645,344), (648,7), (662,6), (664,7), (666,7), (668,7), (670,7), (671,7), (811,264), (886,7), (889,7), (892,7), (903,7)
X(8) = NAGEL POINT
Trilinears (b + c - a)/a : (c + a - b)/b : (a + b - c)/c= csc2(A/2) : csc2(B/2) : csc2(C/2)
Barycentrics b + c - a : c + a - b : a + b - c
Let A'B'C' be the points in which the A-excircle meets BC, the B-excircle meets CA, and the C-excircle meets AB, respectively. The lines AA', BB', CC' concur in X(8). Another construction of A' is to start at A and trace around ABC half its perimeter, and similarly for B' and C'. Also, X(8) is the incenter of the anticomplementary triangle.
If you have The Geometer's Sketchpad, you can view Nagel point.
X(8) lies on these lines:
1,2 3,100 4,72 5,1389 6,594 7,65 9,346 11,1320 19,1891 20,40 21,55 29,219 31,987 33,1039 34,1041 35,993 37,941 38,986 56,404 57,1219 58,996 76,668 79,758 80,149 81,1010 101,1311 140,1483 144,516 171,1468 175,1270 176,1271 177,556 178,236 181,959 190,528 192,256 193,894 194,730 197,1603 210,312 213,981 220,294 221,651 224,914 238,983 253,307 274,1002 277,1280 278,1257 291,330 314,1264 315,760 326,1442 344,480 348,664 392,1000 405,943 406,1061 442,495 443,942 474,999 475,1063 491,1267 595,1724 599,1086 631,1385 643,1098 726,1278 860,1068 908,946 961,1460 1015,1574 1016,1083 1034,1895 1036,1183 1124,1377 1211,1834 1281,1282 1317,1388 1335,1378 1500,1573 1672,1680 1673,1681 1674,1679 1675,1679 1857,1896
X(8) is the {X(69),X(75)}-harmonic conjugate of X(7). For a list of other harmonic conjugates of X(8), click More at the top of this page.
X(8) = reflection of X(I) in X(J) for these (I,J): (1,10), (4,355), (20,40), (100,1145), (145,1), (149,80), (192,984), (390,9), (944,2), (962,4), (1320,11), (1482,5), (1483,140)
X(8) = isogonal conjugate of X(56)
X(8) = isotomic conjugate of X(7)
X(8) = cyclocevian conjugate of X(189)
X(8) = complement of X(145)
X(8) = anticomplement of X(1)
X(8) = anticomplementary conjugate of X(8)
X(8) = X(I)-Ceva conjugate of X(J) for these (I,J): (69,329), (72,2), (312,346), (314,312), (333,9)
X(8) = X(I)-cross conjugate of X(J) for these (I,J):
(1,280), (9,2), (10,318), (11,522), (55,281), (72,78), (200,346), (210,9), (219,345), (497,7), (521,100)
X(8) = cevapoint of X(I) and X(J) for these (I,J):
(1,40), (2,144), (6,197), (9,200), (10,72), (11,522), (55,219), (175,176)
X(8) = crosspoint of X(I) and X(J) for these (I,J): (75,312), (314,333)
X(8) = crosssum of X(I) and X(J) for these (I,J):
(1,978), (31,604), (57,1423), (667,1015), (1042,1410), (1400,1402)
X(8) = crossdifference of any two points on line X(649)X(854)
X(8) = X(1)-aleph conjugate of X(1050)
X(8) = X(I)-beth conjugate of X(J) for these (I,J): (8,1), (341,341), (643,3), (668,8), (1043,8)
X(9) = MITTENPUNKT
Trilinears b + c - a : c + a - b : a + b - c= cot(A/2) : cot(B/2) : cot(C/2)
Barycentrics a(b + c - a) : b(c + a - b) : c(a + b - c)
X(9) is the symmedian point of the excentral triangle.
If you have The Geometer's Sketchpad, you can view Mittenpunkt.
X(9) lies on these lines:
1,6 2,7 3,84 4,10 5,1729 8,346 21,41 31,612 32,987 33,212 34,201 35,90 38,614 39,978 42,941 43,256 46,79 48,101 55,200 56,1696 58,975 100,1005 164,168 165,910 171,1707 173,177 192,239 223,1073 228,1011 241,269 261,645 294,1253 312,314 318,1896 321,1751 342,653 348,738 364,366 374,517 393,1785 440,1211 478,1038 498,920 522,657 604,1420 607,1039 608,1041 609,1333 644,1320 654,1639 750,896 943,1802 986,1722 991,1818 1088,1223 1125,1732 1174,1621 1249,1712 1377,1703 1378,1702 1479,1752 1571,1574 1572,1573 1678,1705 1679,1704 1680,1701 1681,1700
X(9) is the {X(44),X(45)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(9), click More at the top of this page. X(9) is the internal center of similitude of the Bevan circle and Spieker circle; the external center is X(1706).
X(9) = midpoint of X(I) and X(J) for these (I,J): (7,144), (8,390)
X(9) = reflection of X(I) in X(J) for these (I,J): (1,1001), (7,142)
X(9) = isogonal conjugate of X(57)
X(9) = isotomic conjugate of X(85)
X(9) = complement of X(7)
X(9) = anticomplement of X(142)
X(9) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,1), (8,200), (21,55), (63,40), (190,522), (312,78), (318,33), (329, 1490), (333,8)
X(9) = cevapoint of X(I) and X(J) for these (I,J): (1,165), (6,198), (31,205), (37,71), (41,212), (55,220)
X(9) = X(I)-cross conjugate of X(J) for these (I,J):
(6,282), (37,281), (41,33), (55,1), (71,219), (210,8), (212,78), (220,200)
X(9) = crosspoint of X(I) and X(J) for these (I,J): (2,8), (21,333), (63,271), (312,318)
X(9) = crosssum of X(I) and X(J) for these (I,J): (6,56), (19,208), (65,1400), (244,649), (603,604), (1418,1475)
X(9) = crossdifference of any two points on line X(513)X(663)
X(9) = X(I)-Hirst inverse of X(J) for these (I,J): (1, 518), (192,239)
X(9) = X(I)-aleph conjugate of X(J) for these (I,J):
(1,43), (2,9), (9,170), (188,165), (190,1018), (366,1), (507,361), (508,57), (509,978)
X(9) = X(I)-beth conjugate of X(J) for these (I,J):
(9,6), (190,6), (346,346), (644,9), (645,75)
X(10) = SPIEKER CENTER
Trilinears bc(b + c) : ca(c + a) : ab(a + b)Barycentrics b + c : c + a : a + b
The Spieker circle is the incircle of the medial triangle; its center, X(10), is the centroid of the perimeter of ABC.
If you have The Geometer's Sketchpad, you can view Spieker center.
X(10) lies on these lines:
1,2 3,197 4,9 5,517 6,1377 11,121 12,65 20,165 21,35 28,1891 29,1794 31,964 33,406 34,475 36,404 37,594 38,596 39,730 44,752 46,63 55,405 56,474 57,388 58,171 69,969 75,76 81,1224 82,83 86,319 87,979 92,1838 98,101 106,1222 116,120 117,123 119,124 140,214 141,142 150,1282 153,1768 158,318 182,1678 190,671 191,267 201,225 219,965 227,1214 235,1902 255,1771 257,1581 261,1326 274,291 307,1254 321,756 348,1323 391,1743 407,1867 427,1829 429,1824 480,954 485,1686 486,1685 497,1697 514,764 535,1155 537,1086 626,760 631,944 632,1483 750,1150 774,1736 775,801 846,1247 894,1046 908,994 962,1695 1018,1334 1074,1735 1146,1212 1482,1656 1587,1703 1588,1702 1762,1782 1828,1883 1900,1904
X(10) is the {X(1),X(2)}-harmonic conjugate of X(1125). For a list of other harmonic conjugates of X(10), click More at the top of this page. X(10) is the internal center of similitude of the Apollonius and nine-points circles.
X(10) = midpoint of X(I) and X(J) for these (I,J): (1,8), (3,355), (4,40), (65,72), (80,100)
X(10) = reflection of X(I) in X(J) for these (I,J): (1,1125), (551,2), (946,5), (1385,140)
X(10) = isogonal conjugate of X(58)
X(10) = isotomic conjugate of X(86)
X(10) = inverse-in-circumcircle of X(1324)
X(10) = complement of X(1)
X(10) = anticomplement of X(1125)
X(10) = complementary conjugate of X(10)
X(10) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,37), (8,72), (75,321), (80,519), (100,522), (313,306)
X(10) = cevapoint of X(I) and X(J) for these (I,J): (1,191), (6,199), (12,201), (37,210), (42,71), (65,227)
X(10) = X(I)-cross conjugate of X(J) for these (I,J): (37,226), (71,306), (191,502), (201,72)
X(10) = crosspoint of X(I) and X(J) for these (I,J): (2,75), (8,318)
X(10) = crosssum of X(I) and X(J) for these (I,J): (6,31), (56,603)
X(10) = crossdifference of any two points on line X(649)X(834)
X(10) = X(I)-beth conjugate of X(J) for these (I,J): (8,10), (10,65), (100,73), (318,225), (643,35), (668,349)
X(11) = FEUERBACH POINT
Trilinears 1 - cos(B - C) : 1 - cos(C - A) : 1 - cos(A - B)= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin2(B/2 - C/2)
= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc(b + c - a)(b - c)2
Barycentrics a(1 - cos(B - C)) : b(1 - cos(C - A)) : c(1 - cos(A - B))
= h(a,b,c) : h(b,c,a) : h(c,a,b), where h(A,B,C) = (b + c - a)(b - c)2
X(11) is the point of tangency of the nine-point circle and the incircle. The nine-point circle is the circumcenter of the medial triangle, as well as the orthic triangle. Feuerbach's famous theorem states that the nine-point circle is tangent to the incircle and the three excircles.
If you have The Geometer's Sketchpad, you can view Feuerbach point.
X(11) is the {X(1),X(5)}-harmonic conjugate of X(12). For a list of other harmonic conjugates of X(11), click More at the top of this page.
X(11) lies on these lines:
1,5 2,55 3,499 4,56 7,658 8,1320 10,121 13,202 14,203 28,1852 30,36 33,427 34,235 35,140 57,1360 65,117 68,1069 110,215 113,942 115,1015 118,226 124,1364 182,1848 133,1838 153,388 212,748 214,442 244,867 278,1857 325,350 381,999 403,1870 429,1104 485,1124 486,1335 498,1656 515,1319 516,1155 517,1737 518,908 523,1090 613,1352 650,1566 774,1393 944,1388 962,1788 971,1538 1012,1470 1040,1368 1111,1358 1146,1639 1193,1834 1427,1856 1428,1503 1455,1877 1500,1506 1697,1698
X(11) = midpoint of X(I) and X(J) for these (I,J): (1,80), (4,104), (5,1484), (100,149)
X(11) = reflection of X(I) in X(J) for these (I,J): (1,1387), (119,5), (214,1125), (1145,10), (1317,1), (1537,946)
X(11) = isogonal conjugate of X(59)
X(11) = inverse-in-Furhmann-circle of X(1837)
X(11) = complement of X(100)
X(11) = anticomplement of X(3035)
X(11) = complementary conjugate of X(513)
X(11) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,523), (4,513), (7,514), (8,522), (262,1491)
X(11) = crosspoint of X(I) and X(J) for these (I,J): (7,514), (8,522)
X(11) = crosssum of X(I) and X(J) for these (I,J): (6,692), (55,101), (56,109), (1381,1382), (1397,1415)
X(11) = crossdifference of any two points on line X(101)X(109)
X(11) = X(I)-beth conjugate of X(J) for these (I,J): (11,244), (522,11), (693,11)
X(12) = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11)
Trilinears 1 + cos(B - C) : 1 + cos(C - A) : 1 + cos(A - B)= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos2(B/2 - C/2)
= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc(b + c)2/(b + c - a)
Barycentrics a(1 + cos(B - C)) : b(1 + cos(C - A)) : c(1 + cos(A - B))
= h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = (b + c)2/(b + c - a)
Let A'B'C' be the touch points of the nine-point circle with the A-, B-, C- excircles, respectively. The lines AA', BB', CC' concur in X(12).
X(12) lies on these lines:
1,5 2,56 3,498 4,55 7,1268 10,65 17,203 18,202 30,35 33,235 34,427 36,140 37,225 38,1393 40,1836 42,1834 54,215 57,1224 63,1454 71,1901 79,484 85,120 108,451 115,1500 116,1362 117,1364 121,1357 123,1359 124,1361 125,1425 141,1469 171,1399 172,230 201,756 208,1360 221,1853 228,407 281,1118 313,349 354,1210 377,1259 381,1479 431,1824 443,1466 474,1470 485,1124 486,1124 499,999 603,750 611,1352 908,960 942,1209 946,1532 968,1904 1015,1506 1038,1368 1091,1109 1125,1319 1213,1400 1452,1892 1594,1870 1697,1699 1861,1887 1877,1883
X(12) is the {X(1),X(5)}-harmonic conjugate of X(11). For a list of other harmonic conjugates of X(12), click More at the top of this page.
If you have The Geometer's Sketchpad, you can view X(12).
X(12) = isogonal conjugate of X(60)
X(12) = isotomic conjugate of X(261)
X(12) = complement of X(2975)
X(12) = X(10)-Ceva conjugate of X(201)
X(12) = crosssum of X(58) and X(1437)
X(12) = X(I)-beth conjugate of X(J) for these (I,J): (10,12), (1089,1089)
X(13) = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)
Trilinears csc(A + π/3) : csc(B + π/3) : csc(C + π/3)= sec(A - π/6) : sec(B - π/6) : sec(C - π/6)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a4 - 2(b2 - c2)2 + a2(b2 + c2 + 4*sqrt(3)*Area(ABC))
Construct the equilateral triangle BA'C having base BC and vertex A' on the negative side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA',BB',CC' concur in X(13). If each of the angles A, B, C is < 2*π/3, then X(13) is the only center X for which the angles BXC, CXA, AXB are equal, and X(13) minimizes the sum |AX| + |BX| + |CX|. The antipedal triangle of X(13) is equilateral.
If you have The Geometer's Sketchpad, you can view these sketches:
1st isogonic center
Kiepert Hyperbola, showing X(13) and X(14) on the hyperbola, with midpoint X(115).
Evans Conic, passing through X(13), X(14), X(15), X(16), X(17), X(18), X(3070), X(3071).
X(3054), center of the Evans Conic and 19 other triangle centers.
The Evans conic is introduced in
Evans, Lawrence S., "A Conic Through Six Triangle Centers," Forum Geometricorum 2 (2002) 89-92.
X(13) lies on these lines:
2,16 3,17 4,61 5,18 6,14 11,202 15,30 76,299 80,1251 98,1080 99,303 148,617 203,1478 226,1081 262,383 275,472 298,532 484,1277 531,671 533,621 634,635
X(13) is the {X(6),X(381)}-harmonic conjugate of X(14). For a list of other harmonic conjugates of X(13), click More at the top of this page.
X(13) = reflection of X(I) in X(J) for these (I,J): (14,115), (15,396), (99,619), (298,623), (616,618)
X(13) = isogonal conjugate of X(15)
X(13) = isotomic conjugate of X(298)
X(13) = inverse-in-orthocentroidal-circle of X(14)
X(13) = complement of X(616)
X(13) = anticomplement of X(618)
X(13) = cevapoint of X(15) and X(62)
X(13) = X(I)-cross conjugate of X(J) for these (I,J): (15,18), (30,14), (396,2)
X(14) = 2nd ISOGONIC CENTER
Trilinears csc(A - π/3) : csc(B - π/3) : csc(C - π/3)= sec(A + π/6) : sec(B + π/6) : sec(C + π/6)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a4 - 2(b2 - c2)2 + a2(b2 + c2 - 4*sqrt(3)*Area(ABC))
Construct the equilateral triangle BA'C having base BC and vertex A' on the positive side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA',BB',CC' concur in X(14). The antipedal triangle of X(14) is equilateral.
If you have The Geometer's Sketchpad, you can view 2nd isogonic center
X(14) lies on these lines:
2,15 3,18 4,62 5,17 6,13 11,203 16,30 76,298 98,383 99,302 148,616 202,1478 226,554 262,1080 275,473 299,533 397,546 484,1276 530,671 532,622 633,636
X(14) is the {X(6),X(381)}-harmonic conjugate of X(13). For a list of other harmonic conjugates of X(14), click More at the top of this page.
X(14) = reflection of X(I) in X(J) for these (I,J): (13,115), (16,395), (99,618), (299,624), (617,619)
X(14) = isogonal conjugate of X(16)
X(14) = isotomic conjugate of X(299)
X(14) = inverse-in-orthocentroidal-circle of X(13)
X(14) = complement of X(617)
X(14) = anticomplement of X(619)
X(14) = cevapoint of X(16) and X(61)
X(14) = X(I)-cross conjugate of X(J) for these (I,J): (16,17), (30,13), (395,2)
X(15) = 1st ISODYNAMIC POINT
Trilinears sin(A + π/3) : sin(B + π/3) : sin(C + π/3)= cos(A - π/6) : cos(B - π/6) : cos(C - π/6)
Barycentrics a sin(A + π/3) : b sin(B + π/3) : c sin(C + π/3)
Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through one vertex and both isodynamic points. (This configuration, based on the cevians of X(1), generalizes to arbitrary cevians; see TCCT, p. 98, problem 8.)
The pedal triangle of X(15) is equilateral.
If you have The Geometer's Sketchpad, you can view 1st isodynamic point and X(15)&X(16), with Brocard axis and Lemoine axis.
X(15) lies on these lines:
1,1251 2,14 3,6 4,17 13,30 18,140 35,1250 36,202 55,203 298,533 303,316 395,549 397,550 532,616 628,636
X(15) is the {X(3),X(6)}-harmonic conjugate of X(16). For a list of other harmonic conjugates of X(15), click More at the top of this page.
X(15) = reflection of X(I) in X(J) for these (I,J): (13,396), (16,187), (298,618), (316,624), (621,623)
X(15) = isogonal conjugate of X(13)
X(15) = isotomic conjugate of X(300)
X(15) = inverse-in-circumcircle of X(16)
X(15) = inverse-in-Brocard-circle of X(16)
X(15) = complement of X(621)
X(15) = anticomplement of X(623)
X(15) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,202), (13,62), (74,16)
X(15) = crosspoint of X(I) and X(J) for these (I,J): (13,18), (298,470)
X(15) = crosssum of X(I) and X(J) for these (I,J): (15,62), (532,619)
X(15) = crossdifference of any two points on line X(395)X(523)
X(15) = X(6)-Hirst inverse of X(16)
X(16) = 2nd ISODYNAMIC POINT
Trilinears sin(A - π/3) : sin(B - π/3) : sin(C - π/3)= cos(A + π/6) : cos(B + π/6) : cos(C + π/6)
Barycentrics a sin(A - π/3) : b sin(B - π/3) : c sin(C - π/3)
Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through a vertex and both isodynamic points. The pedal triangle of X(16) is equilateral.
If you have The Geometer's Sketchpad, you can view 2nd isodynamic point.
X(16) lies on these lines:
1,1250 2,13 3,6 4,18 14,30 17,140 36,203 55,202 299,532 302,316 358,1135 396,549 398,550 533,617 627,635
X(16) is the {X(3),X(6)}-harmonic conjugate of X(15). For a list of other harmonic conjugates of X(16), click More at the top of this page.
X(16) = reflection of X(I) in X(J) for these (I,J): (14,395), (15,187), (299,619), (316,623), (622,624)
X(16) = isogonal conjugate of X(14)
X(16) = isotomic conjugate of X(301)
X(16) = inverse-in-circumcircle of X(15)
X(16) = inverse-in-Brocard-circle of X(15)
X(16) = complement of X(622)
X(16) = anticomplement of X(624)
X(16) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,203), (14,61), (74,15)
X(16) = crosspoint of X(I) and X(J) for these (I,J): (14,17), (299,471)
X(16) = crosssum of X(I) and X(J) for these (I,J): (16,61), (533,618)
X(16) = crossdifference of any two points on line X(396)X(523)
X(16) = X(6)-Hirst inverse of X(15)
X(17) = 1st NAPOLEON POINT
Trilinears csc(A + π/6) : csc(B + π/6) : csc(C + π/6)= sec(A - π/3) : sec(B - π/3) : sec(C - π/3)
Barycentrics a csc(A + π/6) : b csc(B + π/6) : c csc(C + π/6)
Let X,Y,Z be the centers of the equilateral triangles in the construction of X(13). The lines AX, BY, CZ concur in X(17).
John Rigby, "Napoleon revisited," Journal of Geometry, 33 (1988) 126-146.
If you have The Geometer's Sketchpad, you can view 1st Napoleon point.
X(17) lies on these lines:
2,62 3,13 4,15 5,14 6,18 12,203 16,140 76,303 83,624 202,499 275,471 299,635 623,633
X(17) is the {X(231),X(1209)}-harmonic conjugate of X(18). For a list of other harmonic conjugates of X(17), click More at the top of this page.
X(17) = reflection of X(627) in X(629)
X(17) = isogonal conjugate of X(61)
X(17) = isotomic conjugate of X(302)
X(17) = complement of X(627)
X(17) = anticomplement of X(629)
X(17) = X(I)-cross conjugate of X(J) for these (I,J): (16,14), (140,18), (397,4)
X(18) = 2nd NAPOLEON POINT
Trilinears csc(A - π/6) : csc(B - π/6) : csc(C - π/6)= sec(A + π/3) : sec(B + π/3) : sec(C + π/3)
Barycentrics a csc(A - π/6) : b csc(B - π/6) : c csc(C - π/6)
Let X,Y,Z be the centers of the equilateral triangles in the construction of X(14). The lines AX, BY, CZ concur in X(18).
If you have The Geometer's Sketchpad, you can view 2nd Napoleon point.
X(18) lies on these lines:
2,61 3,14 4,16 5,13 6,17 12,202 15,140 76,302 83,623 203,499 275,470 298,636 624,634
X(18) is the {X(231),X(1209)}-harmonic conjugate of X(17). For a list of other harmonic conjugates of X(18), click More at the top of this page.
X(18) = reflection of X(628) in X(630)
X(18) = isogonal conjugate of X(62)
X(18) = isotomic conjugate of X(303)
X(18) = complement of X(628)
X(18) = anticomplement of X(630)
X(18) = X(I)-cross conjugate of X(J) for these (I,J): (15,13), (140,17), (398,4)
X(19) = CLAWSON POINT
Trilinears tan A : tan B : tan C= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin 2B + sin 2C - sin 2A
= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/(b2 + c2 - a2)
Barycentrics a tan A : b tan B : c tan C
X(19) is the homothetic center of the orthic and extangents triangles.
If you have The Geometer's Sketchpad, you can view Clawson point.
Further information is available from
Paul Yiu's Website.
Although John Clawson studied this point in 1925, it was studied earlier by Lemoine:
Emile Lemoine, "Quelques questions se rapportant à l'étude des antiparallèles des côtes d'un triangle", Bulletin de la S. M. F., tome 14 (1886), p. 107-128, specifically, on page 114. This article is available online at Numdam.
X(19) lies on these lines:
1,28 2,534 3,1871 4,9 6,34 8,1891 25,33 27,63 31,204 41,1825 44,1828 45,1900 46,579 47,921 53,1846 56,207 57,196 64,1903 81,969 91,920 101,913 102,282 112,759 158,1712 162,897 163,563 208,225 219,517 220,1902 226,1763 232,444 273,653 294,1041 318,1840 379,1441 407,1865 429,1213 560,1910 604,909 672,1851 960,965 1158,1715 1212,1593 1405,1866 1449,1870 1581,1740 1598,1872 1633,1721 1707,1719 1708,1713 1743,1783 1836,1901 1837,1852
X(19) is the {X(607),X(608)}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(19), click More at the top of this page.
X(19) = isogonal conjugate of X(63)
X(19) = isotomic conjugate of X(304)
X(19) = X(I)-Ceva conjugate of X(J) for these (I,J):
(1,204), (4,33), (27,4), (28,25), (57,208), (92,1), (196,207), (278,34)
X(19) = X(I)-cross conjugate of X(J) for these (I,J): (25,34), (31,1)
X(19) = crosspoint of X(I) and X(J) for these (I,J): (4,278), (27,28), (57,84), (92,158)
X(19) = crosssum of X(I) and X(J) for these (I,J): (1,610), (3,219), (9,40), (48,255), (71,72)
X(19) = crossdifference of any two points on line X(521)X(656)
X(19) = X(I)-Hirst inverse of X(J) for these (I,J): (1,240), (4,242)
X(19) = X(I)-aleph conjugate of X(J) for these (I,J): (2,610), (92,19), (508,223), (648,163)
X(19) = X(I)-beth conjugate of X(J) for these (I,J): (9,198), (19,608), (112,604), (281,281), (648,273), (653,19)
Centers 20- 30,
2- 5, 140, 186, 199, 235, 237, 297, 376- 379, 381- 384,
401- 475, 546- 550, 631, 632 (and others) lie on the Euler line.
X(20) = DE LONGCHAMPS POINT
Trilinears cos A - cos B cos C : cos B - cos C cos A : cos C - cos A cos B= sec A - sec B sec C : sec B - sec C sec A : sec C - sec A sec B
Barycentrics tan B + tan C - tan A : tan C + tan A - tan B : tan A + tan B - tan C
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [-3a4 + 2a2(b2 + c2) + (b2 - c2)2]
X(2) is the reflection of X(4) in X(3); also, the orthocenter of the anticomplementary triangle.
If you have The Geometer's Sketchpad, you can view De Longchamps point.
X(20) lies on these lines:
1,7 2,3 8,40 10,165 33,1038 34,1040 35,1478 36,1479 55,388 56,497 57,938 58,387 64,69 68,74 72,144 78,329 97,1217 98,148 99,147 100,153 101,152 103,150 104,149 109,151 110,146 145,517 155,323 185,193 190,1265 243,1118 254,1300 346,1766 371,1587 372,1588 391,573 393,577 394,1032 485,1131 486,1132 487,638 488,637 616,633 617,635 621,627 622,628 936,1750 999,1058 1062,1870 1074,1838 1076,1785 1125,1699 1147,1614 1155,1788 1204,1899 1440,1804 1610,1633
X(20) is the {X(3),X(4)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(20), click More at the top of this page.
X(20) = reflection of X(I) in X(J) for these (I,J): (2,376), (3,550), (4,3), (5,548), (8,40), (69,1350), (145,944), (146,110), (147,99), (148,98), (149,104), (150,103), (151,109), (152,101), (153,100), (382,5), (962,1)
X(20) = isogonal conjugate of X(64)
X(20) = isotomic conjugate of X(253)
X(20) = cyclocevian conjugate of X(1032)
X(20) = inverse-in-circumcircle of X(2071)
X(20) = inverse-in-orthocentroidal-circle of X(3091)
X(20) = complement of X(3146)
X(20) = anticomplement of X(4)
X(20) = anticomplementary conjugate of X(4)
X(20) = X(I)-Ceva conjugate of X(J) for these (I,J): (69,2), (489,487), (490,488)
X(20) = crosssum of X(1) and X(1044)
X(20) = crossdifference of any two points on line X(647)X(657)
X(20) = X(I)-aleph conjugate of X(J) for these (I,J): (8,191), (9,1045), (188,1046), (333,2), (1043,20)
X(20) = X(I)-beth conjugate of X(J) for these (I,J): (664,20), (1043,280)
X(21) = SCHIFFLER POINT
Trilinears 1/(cos B + cos C) : 1/(cos C + cos A) : 1/(cos A + cos B)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b + c)
Barycentrics a/(cos B + cos C) : b/(cos C + cos A) : c/(cos A + cos B)
Write I for the incenter; the Euler lines of the four triangles IBC, ICA, IAB, and ABC concur in X(21).
If you have The Geometer's Sketchpad, you can view Schiffler point.
Lev Emelyanov and Tatiana Emelyanova, A note on the Schiffler point, Forum Geometricorum 3 (2003) pages 113-116.
The name of this point honors Kurt Schiffler.
X(21) lies on these lines:
1,31 2,3 6,941 7,56 8,55 9,41 10,35 32,981 36,79 37,172 51,970 60,960 72,943 75,272 77,1394 84,285 90,224 99,105 104,110 107,1295 144,954 145,956 238,256 243,1896 261,314 268,280 270,1172 286,1441 294,1212 332,1036 385,1655 386,1724 517,1389 572,1765 600,1698 612,989 614,988 643,1320 644,1334 662,1156 741,932 748,978 884,885 915,925 961,1402 976,983 1030,1213 1038,1041 1039,1040 1060,1063 1061,1062 1214,1396 1254,1758 1319,1408 1412,1420
X(21) is the {X(2),X(3)}-harmonic conjugate of X(404). For a list of other harmonic conjugates of X(21), click More at the top of this page.
X(21) = midpoint of X(1) and X(191)
X(21) = isogonal conjugate of X(65)
X(21) = isotomic conjugate of X(1441)
X(21) = inverse-in-circumcircle of X(1325)
X(21) = anticomplement of X(442)
X(21) = X(I)-Ceva conjugate of X(J) for these (I,J): (86,81), (261,333)
X(21) = cevapoint of X(I) and X(J) for these (I,J): (1,3), (9,55)
X(21) = X(I)-cross conjugate of X(J) for these (I,J):
(1,29), (3,283), (9,333), (55,284), (58,285), (284,81), (522,100)
X(21) = crosspoint of X(86) and X(333)
X(21) = crosssum of X(I) and X(J) for these (I,J): (1,1046), (42,1400), (1254,1425), (1402,1409)
X(21) = crossdifference of any two points on line X(647)X(661)
X(21) = X(I)-Hirst inverse of X(J) for these (I,J): (2,448), (3,416), (4,425)
X(21) = X(I)-beth conjugate of X(J) for these (I,J): (21,58), (99,21), (643,21), (1043,1043), (1098,21)
X(22) = EXETER POINT
Trilinears a(b4 + c4 - a4) : b(c4 + a4 - b4) : c(a4 + b4 - c4)Barycentrics a2(b4 + c4 - a4) : b2(c4 + a4 - b4) : c2(a4 + b4 - c4)
Barycentrics sin 2A - tan ω : sin 2B - tan ω : sin 2C - tan ω (M. Iliev, 5/13/07)
X(22) is the perspector of the circummedial triangle and the tangential triangle; also X(22) = X(55)-of-the-tangential-triangle if ABC is acute. See the note just before X(1601) for a generalization.
If you have The Geometer's Sketchpad, you can view Exeter point.
X(22) lies on these lines:
2,3 6,251 32,1194 35,612 36,614 51,182 56,977 69,159 76,1799 98,925 99,305 100,197 110,154 155,1614 157,183 160,325 161,343 184,511 187,1196 232,577 264,1629 347,1617 675,1305 991,1790 1184,1627 1294,1302 1486,1621 1602,1626
X(22) is the {X(3),X(25)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(22), click More at the top of this page.
X(22) = reflection of X(378) in X(3)
X(22) = isogonal conjugate of X(66)
X(22) = inverse-in-circumcircle of X(858)
X(22) = anticomplement of X(427)
X(22) = X(76)-Ceva conjugate of X(6)
X(22) = cevapoint of X(3) and X(159)
X(22) = crosspoint of X(99) and X(250)
X(22) = crosssum of X(125) and X(512)
X(22) = crossdifference of any two points on line X(647)X(826)
X(22) = X(I)-beth conjugate of X(J) for these (I,J): (643,345), (833,22)
X(23) = FAR-OUT POINT
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b4 + c4 - a4 - b2c2]Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics 2 sin 2A - 3 tan ω : 2 sin 2B - 3 tan ω : 2 sin 2C - 3 tan ω (M. Iliev, 5/13/07)
X(23) is the inverse-in-circumcircle of X(2).
If you have The Geometer's Sketchpad, you can view Far-out point.
X(23) lies on these lines:
2,3 6,353 51,575 52,1614 94,98 105,1290 110,323 111,187 143,1199 159,193 184,576 232,250 251,1194 324,1629 385,523 477,1302 895,1177 1196,1627 1297,1804
X(23) is the {X(22),X(25)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(23), click More at the top of this page.
X(23) = reflection of X(I) in X(J) for these (I,J): (110,1495), (323,110), (691,187), (858,468)
X(23) = isogonal conjugate of X(67)
X(23) = inverse-in-circumcircle of X(2)
X(23) = anticomplement of X(858)
X(23) = crosspoint of X(111) and X(251)
X(23) = crosssum of X(I) and X(J) for these (I,J): (125,690), (141,524)
X(23) = crossdifference of any two points on line X(39)X(647)
X(24) = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE
Trilinears sec A cos 2A : sec B cos 2B : sec C cos 2C= sec A - 2 cos A : sec B - 2 cos B : sec C - 2 cos C
Barycentrics tan A cos 2A : tan B cos 2B : tan C cos 2C
= tan A - sin 2A : tan A - sin 2B : tan C - sin 2C
Constructed as indicated by the name; also X(24) = X(56)-of-the-tangential triangle if ABC is acute.
If you have The Geometer's Sketchpad, you can view X(24).
X(24) lies on these lines:
1,1061 2,3 6,54 32,232 33,35 34,36 49,568 51,578 52,1147 56,1870 64,74 96,847 98,1289 107,1093 108,915 110,155 154,1181 182,1843 183,1235 184,389 185,1495 242,1602 254,393 264,1078 511,1092 573,1474 602,1395 944,1610 1063,1775 1112,1511 1192,1511 1324,1603 1385,1829
X(24) is the {X(3),X(4)}-harmonic conjugate of X(378). For a list of other harmonic conjugates of X(24), click More at the top of this page.
X(24) = reflection of X(4) in X(235)
X(24) = isogonal conjugate of X(68)
X(24) = inverse-in-circumcircle of X(403)
X(24) = X(249)-Ceva conjugate of X(112)
X(24) = X(52)-cross conjugate of X(4)
X(24) = crosspoint of X(107) and X(250)
X(24) = crosssum of X(I) and X(J) for these (I,J): (6,161), (125,520), (637,638)
X(24) = X(4)-Hirst inverse of X(421)
X(25) = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES
Trilinears sin A tan A : sin B tan B : sin C tan C = cos A - sec A : cos B - sec B : cos C - sec C= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(b2 + c2 - a2)
Barycentrics sin 2A - 2 tan A : sin 2B - 2 tan B : sin 2C - 2 tan C
= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/(b2 + c2 - a2)
Constructed as indicated by the name; also X(25) is the pole of the orthic axis (the line having trilinear coefficients cos A : cos B : cos C) with respect to the circumcircle. Also, X(25) is X(57)-of-the-tangential triangle.
If you have The Geometer's Sketchpad, you can view X(25).
X(25) lies on these lines:
1,1036 2,3 6,51 19,33 31,608 32,1184 34,56 35,1900 36,1878 40,1902 41,42 52,155 53,157 57,1473 58,967 64,1192 65,1452 76,1241 92,242 98,107 100,1862 105,108 110,1112 111,112 114,135 125,1853 132,136 143,156 183,264 185,1498 221,1425 225,1842 226,1892 262,275 273,1447 286,1218 317,325 339,1289 343,1352 371,493 372,494 389,1181 393,1033 394,511 669,878 692,913 694,1613 842,1304 847,1179 941,1172 958,1891 999,1870 1001,1848 1073,1297 1096,1402 1235,1239 1300,1302 1324,1785 1376,1861 1470,1877 1503,1619 1604,1863 1631,1826 1726,1736 1730,1754
X(25) is the {X(5),X(26)}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(25), click More at the top of this page.
X(25) = reflection of X(I) in X(J) for these (I,J): (4,1596), (1370,1368)
X(25) = isogonal conjugate of X(69)
X(25) = isotomic conjugate of X(305)
X(25) = inverse-in-circumcircle of X(468)
X(25) = inverse-in-orthocentroidal-circle of X(427)
X(25) = complement of X(1370)
X(25) = anticomplement of X(1368)
X(25) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,6), (28,19), (250,112)
X(25) = X(32)-cross conjugate of X(6)
X(25) = crosspoint of X(I) and X(J) for these (I,J): (4,393), (6,64), (19,34), (112,250)
X(25) = crosssum of X(I) and X(J) for these (I,J): (2,20), (3,394), (6,159), (8,329), (63,78), (72,306), (125,525)
X(25) = crossdifference of any two points on line X(441)X(525)
X(25) = X(I)-Hirst inverse of X(J) for these (I,J): (4,419), (6,232)
X(25) = X(I)-beth conjugate of X(J) for these (I,J): (33,33), (108,25), (162,278)
X(26) = CIRCUMCENTER OF THE TANGENTIAL TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b2cos 2B + c2cos 2C - a2cos 2A]Trilinears g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (J2 - 3) cos A + 4 cos B cos C, where J is as at X(1113)
Barycentrics h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = a2(b2cos 2B + c2cos 2C - a2cos 2A)
If you have The Geometer's Sketchpad, you can view X(26).
Theorems involving X(26), published in 1889 by A. Gob, are discussed in
Roger A. Johnson, Advanced Euclidean Geometry, Dover, 1960, 259-260.
X(26) lies on these lines: 2,3 6,143 52,184 68,161 98,1286 154,155 206,511 1605,1607 1606,1608
X(26) is the {X(154),X(155)}-harmonic conjugate of X(156). For a list of other harmonic conjugates of X(26), click More at the top of this page.
X(26) = reflection of X(155) in X(156)
X(26) = isogonal conjugate of X(70)
X(26) = inverse-in-circumcircle of X(2072)
X(26) = crosssum of X(125) and X(924)
X(27) = CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER
Trilinears (sec A)/(b + c) : (sec B)/(c + a) : (sec C)/(a + b)Barycentrics (tan A)/(b + c) : (tan B)/(c + a) : (tan C)/(a + b)
If you have The Geometer's Sketchpad, you can view X(27).
X(27) lies on these lines:
2,3 6,1246 7,81 19,63 57,273 58,270 84,1896 86,1474 103,107 110,917 226,284 239,1829 243,1859 295,335 306,1043 393,967 579,1751 648,903 662,913 1014,1440 1088,1434 1268,1796 1719,1733 1730,1746 1770,1780
X(27) is the {X(2),X(4)}-harmonic conjugate of X(469). For a list of other harmonic conjugates of X(27), click More at the top of this page.
X(27) = isogonal conjugate of X(71)
X(27) = isotomic conjugate of X(306)
X(27) = inverse-in-orthocentroidal-circle of X(469)
X(27) = complement of X(3151)
X(27) = anticomplement of X(440)
X(27) = X(286)-Ceva conjugate of X(29)
X(27) = cevapoint of X(I) and X(J) for these (I,J): (4,19), (57,278)
X(27) = X(I)-cross conjugate of X(J) for these (I,J): (4,286), (19,28), (57,81), (58,86)
X(27) = crossdifference of any two points on line X(647)X(810)
X(27) = X(I)-Hirst inverse of X(J) for these (I,J): (2,447), (4,423)
X(27) = X(I)-beth conjugate of X(J) for these (I,J): (648,27), (923,27)
X(28) = CEVAPOINT OF X(19) AND X(25)
Trilinears (tan A)/(b + c) : (tan B)/(c + a) : (tan C)/(a + b)Barycentrics (sin A tan A)/(b + c) : (sin B tan B)/(c + a) : (sin C tan C)/(a + b)
If you have The Geometer's Sketchpad, you can view X(28).
X(28) lies on these lines:
1,19 2,3 10,1891 11,1852 33,975 34,57 35,1869 36,1838 46,1780 54,1243 56,278 60,81 65,1175 72,1257 88,162 104,107 105,112 108,225 110,915 142,1890 228,943 242,261 272,273 279,1014 281,958 291,1783 501,1831 579,1724 580,1730 607,1002 608,959 614,1472 956,1219 957,1191 961,1169 1104,1333 1125,1848 1155,1888 1170,1876 1178,1432 1224,1826 1255,1824 1295,1301 1385,1871 1412,1422 1633,1770 1710,1725
X(28) is the {X(27),X(29)}-harmonic conjugate of X(4). For a list of other harmonic conjugates of X(28), click More at the top of this page.
X(28) = isogonal conjugate of X(72)
X(28) = X(I)-Ceva conjugate of X(J) for these (I,J): (270,58), (286,81)
X(28) = cevapoint of X(I) and X(J) for these (I,J): (19,25), (34,56)
X(28) = X(I)-cross conjugate of X(J) for these (I,J): (19,27), (58,58)
X(28) = crossdifference of any two points on line X(647)X(656)
X(28) = X(4)-Hirst inverse of X(422)
X(28) = X(I)-beth conjugate of X(J) for these (I,J): (29,29), (107,28), (162,28), (270,28)
X(29) = CEVAPOINT OF INCENTER AND ORTHOCENTER
Trilinears (sec A)/(cos B + cos C) : (sec B)/(cos C + cos A) : (sec C)/(cos A + cos B)Barycentrics (tan A)/(cos B + cos C) : (tan B)/(cos C + cos A) : (tan C)/(cos A + cos B)
If you have The Geometer's Sketchpad, you can view X(29).
X(29) lies on these lines:
1,92 2,3 8,219 10,1794 33,78 34,77 58,162 65,296 81,189 102,107 112,1311 226,951 242,257 270,283 284,950 314,1039 388,1037 392,1871 497,1036 515,947 648,1121 662,1800 758,1844 894,1868 960,1859 1056,1059 1057,1058 1125,1838 1220,1474 1737,1780 1807,1897 1842,1848
X(29) is the {X(3),X(4)}-harmonic conjugate of X(412). For a list of other harmonic conjugates of X(29), click More at the top of this page.
X(29) = isogonal conjugate of X(73)
X(29) = isotomic conjugate of X(307)
X(29) = complement of X(3153)
X(29) = X(286)-Ceva conjugate of X(27)
X(29) = cevapoint of X(I) and X(J) for these (I,J): (1,4), (33,281)
X(29) = X(I)-cross conjugate of X(J) for these (I,J): (1,21), (284,333), (497,314)
X(29) = crosssum of X(I) and X(J) for these (I,J): (1,1047), (228,1409)
X(29) = crossdifference of any two points on line X(647)X(822)
X(29) = X(4)-Hirst inverse of X(415)
X(29) = X(I)-beth conjugate of X(J) for these (I,J): (29,28), (811,29)
X(30) = EULER INFINITY POINT
Trilinears cos A - 2 cos B cos C : cos B - 2 cos C cos A : cos C - 2 cos A cos B= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[2a4 - (b2 - c2)2 - a2(b2 + c2)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2a4 - (b2 - c2)2 - a2(b2 + c2)
X(30) is the point of intersection of the Euler line and the line at infinity. Thus, each of the 41 lines listed below is parallel to the Euler line.
If you have The Geometer's Sketchpad, you can view Euler Infinity Point.
X(30) lies on these lines:
1,79 2,3 11,36 12,35 13,15 14,16 33,1060 34,1062 40,191 46,1837 49,1614 52,185 53,577 55,495 56,496 58,1834 61,397 62,398 64,68 65,1770 74,265 80,484 98,671 99,316 104,1290 110,477 113,1495 115,187 143,389 146,323 148,385 155,1498 156,1147 182,597 262,598 284,1901 298,616 299,617 340,1494 390,1056 485,1151 486,1152 489,638 490,637 497,999 511,512 551,946 553,942 582,1724 599,1350 618,623 619,624 620,625 841,1302 935,1297 944,962 1043,1330 1141,1157 1155,1737 1294,1304 1319,1387 1351,1353 1465,1877 1838,1852
X(30) = orthopoint of X(523)
X(30) = isogonal conjugate of X(74)
X(30) = isotomic conjugate of X(1494)
X(30) = anticomplementary conjugate of X(146)
X(30) = complementary conjugate of X(113)
X(30) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,113), (265,5), (476,523)
X(30) = cevapoint of X(3) and X(399)
X(30) = crosspoint of X(I) and X(J) for these (I,J): (13,14), (94,264)
X(30) = crosssum of X(I) and X(J) for these (I,J): (15,16), (50,184)
X(30) = crossdifference of any two points on line X(6)X(647)
X(31) = 2nd POWER POINT
Trilinears a2 : b2 : c2= 1 - cos 2A : 1 - cos 2B : 1 - cos 2C
Barycentrics a3 : b3 : c3
If you have The Geometer's Sketchpad, you can view X(31) (1), X(31) (2), X(31) (3).
X(31) lies on these lines:
1,21 2,171 3,601 6,42 8,987 9,612 10,964 19,204 25,608 32,41 35,386 36,995 40,580 43,100 44,210 48,560 51,181 56,154 57,105 65,1104 72,976 75,82 76,734 91,1087 92,162 99,715 101,609 110,593 163,923 184,604 237,904 404,978 561,722 649,884 669,875 701,789 743,825 745,827 759,994 775,1097 937,1103 940,1001 999,1149 1139,1140
X(31) is the {X(1),X(63)}-harmonic conjugate of X(38). For a list of other harmonic conjugates of X(21), click More at the top of this page.
X(31) = isogonal conjugate of X(75)
X(31) = isotomic conjugate of X(561)
X(31) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,48), (6,41), (9,205), (58,6), (82,1)
X(31) = X(213)-cross conjugate of X(6)
X(31) = crosspoint of X(I) and X(J) for these (I,J): (1,19), (6,56)
X(31) = crosssum of X(I) and X(J) for these (I,J): (1,63), (2,8), (7,347), (10,321), (239,1281), (244,514), (307,1441), (523,1086), (693,1111)
X(31) = crossdifference of any two points on line X(514)X(661)
X(31) = X(1403)-Hirst inverse of X(1428)
X(31) = X(I)-aleph conjugate of X(J) for these (I,J): (82,31), (83,75)
X(31) = X(I)-beth conjugate of X(J) for these (I,J): (21,993), (55,55), (109,31), (110,57), (643,31), (692,31)
X(32) = 3rd POWER POINT
Trilinears a3 : b3 : c3= sin(A - ω) : sin(B - ω) : sin(C - ω)
= sin A + sin(A - 2ω) : sin B + sin(B - 2ω) : sin C + sin(C - 2ω)
= cos A - cos(A - 2ω) : cos B - cos(B - 2ω) : cos C - cos(C - 2ω) (cf., X(39))
Barycentrics a4 : b4 : c4
If you have The Geometer's Sketchpad, you can view X(32).
X(32) lies on these lines:
1,172 2,83 3,6 4,98 5,230 9,987 21,981 24,232 31,41 56,1015 75,746 76,384 81,980 99,194 100,713 101,595 110,729 163,849 184,211 218,906 512,878 538,1003 561,724 590,640 604,1106 615,639 731,825 733,827 910,1104 993,1107
X(32) is the {X(3),X(6)}-harmonic conjugate of X(39). For a list of other harmonic conjugates of X(21), click More at the top of this page.
X(32) = midpoint of X(371) and X(372)
X(32) = reflection of X(315) in X(626)
X(32) = isogonal conjugate of X(76)
X(32) = isotomic conjugate of X(1502)
X(32) = inverse-in-circumcircle of X(1691)
X(32) = inverse-in-Brocard-circle of X(39)
X(32) = inverse-in-1st-Lemoine-circle of X(1692)
X(32) = complement of X(315)
X(32) = anticomplement of X(626)
X(32) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,206), (6,184), (112,512), (251,6)
X(32) = crosspoint of X(I) and X(J) for these (I,J): (2,66), (6,25)
X(32) = crosssum of X(I) and X(J) for these (I,J): (2,69), (6,22), (75,312), (115,826), (311,343), (313,321), (338,850), (339,525), (349,1231), (693,1086), (1229,1233), (1230,1269)
X(32) = crossdifference of any two points on line X(325)X(523)
X(32) = X(184)-Hirst inverse of X(237)
X(32) = X(I)-beth conjugate of X(J) for these (I,J): (41,41), (163,56), (919,32)
X(32) = external center of similitude of circumcircle and Moses circle
X(33) = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES
Trilinears 1 + sec A : 1 + sec B : 1 + sec C = tan A cot(A/2) : tan B cot(B/2) : tan C cot(C/2)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b2 + c2 - a2)
= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec A cos2(A/2)
Barycentrics sin A + tan A : sin B + tan B : sin C + tan C
= h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = tan A cos2(A/2)
If you have The Geometer's Sketchpad, you can view X(33).
X(33) lies on these lines:
1,4 2,1040 5,1062 6,204 7,1041 8,1039 9,212 10,406 11,427 12,235 19,25 20,1038 24,35 28,975 29,78 30,1060 36,378 40,201 42,393 47,90 56,963 57,103 63,1013 64,65 79,1063 80,1061 84,603 112,609 200,281 210,220 222,971 264,350
X(33) is the {X(1),X(4)}-harmonic conjugate of X(34). For a list of other harmonic conjugates, click More at the top of this page.
X(33) = isogonal conjugate of X(77)
X(33) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,19), (29,281), (318,9)
X(33) = X(I)-cross conjugate of X(J) for these (I,J): (41,9), (42,55)
X(33) = crosspoint of X(I) and X(J) for these (I,J): (1,282), (4,281)
X(33) = crosssum of X(I) and X(J) for these (I,J): (1,223), (3,222), (57,1394), (73,1214)
X(33) = crossdifference of any two points on line X(652)X(905)
X(33) = X(33)-beth conjugate of X(25)
X(34) = X(4)-BETH CONJUGATE OF X(4)
Trilinears 1 - sec A : 1 - sec B : 1 - sec C= tan A tan(A/2) : tan B tan(B/2) : tan C tan(C/2)
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(b + c - a)(b2 + c2 - a2)]
= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec A sin2(A/2)
Barycentrics sin A - tan A : sin B - tan B : sin C - tan C
= h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = tan A sin2(A/2)
X(34) is the center of perspective of the orthic triangle and the reflection in the incenter of the intangents triangle.
If you have The Geometer's Sketchpad, you can view X(34) (1) and X(34) (2).
X(34) lies on these lines:
1,4 2,1038 5,1060 6,19 7,1039 8,1041 9,201 10,475 11,235 12,427 20,1040 24,36 25,56 28,57 29,77 30,1062 35,378 40,212 46,47 55,227 79,1061 80,1063 87,242 106,108 196,937 207,1042 222,942 244,1106 331,870 347,452 860,997
X(34) is the {X(1),X(4)}-harmonic conjugate of X(33). For a list of other harmonic conjugates of X(34), click More at the top of this page.
X(34) = isogonal conjugate of X(78)
X(34) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,207), (4,208), (28,56), (273,57), (278,19)
X(34) = X(25)-cross conjugate of X(19)
X(34) = crosssum of X(219) and X(1260)
X(34) = X(56)-Hirst inverse of X(1430)
X(34) = X(I)-beth conjugate of X(J) for these (I,J):
(1,221), (4,4), (28,34), (29,1), (107,158), (108,34), (110,47), (162,34), (811,34)
X(34) = crossdifference of any two points on line X(521)X(652)
X(35) = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36)
Trilinears 1 + 2 cos A : 1 + 2 cos B : 1 + 2 cos C= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2 + bc)
Barycentrics sin A + sin 2A : sin B + sin 2B : sin C + sin 2C
= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 + c2 - a2 + bc)
Let A' be the inverse-in-circumcircle of the A-excenter, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(35).
X(35) lies on these lines:
1,3 4,498 8,993 9,90 10,21 11,140 12,30 22,612 24,33 31,386 34,378 37,267 42,58 43,1011 47,212 71,284 72,191 73,74 79,226 172,187
228,846 255,991 376,388 404,1125
411,516 474,1001 495,550 496,549
497,499 500,1154 595,902 950,1006
968,975 1124,1152
If you have The Geometer's Sketchpad, you can view X(35).
X(35) is the {X(1),X(3)}-harmonic conjugate of X(36). For a list of other harmonic conjugates of X(35), click More at the top of this page.
X(35) = isogonal conjugate of X(79)
X(35) = inverse-in-circumcircle of X(484)
X(35) = X(500)-cross conjugate of X(1)
X(35) = crosssum of X(481) and X(482)
X(35) = X(943)-aleph conjugate of X(35)
X(35) = X(I)-beth conjugate of X(J) for these (I,J): (100,35), (643,10)
X(36) = INVERSE-IN-CIRCUMCIRCLE OF INCENTER
Trilinears 1 - 2 cos A : 1 - 2 cos B : 1 - 2 cos C= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2 - bc)
= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec(A/2) cos(3A/2)
Barycentrics sin A - sin 2A : sin B - sin 2B : sin C - sin 2C
= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 + c2 - a2 - bc)
If you have The Geometer's Sketchpad, you can view X(36).
X(36) is the {X(3),X(56)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(36), click More at the top of this page.
X(36) lies on these lines:
1,3 2,535 4,499 6,609 10,404 11,30 12,140 15,202 16,203 21,79 22,614 24,34 31,995 33,378 39,172 47,602 48,579 54,73 58,60 59,1110 63,997 80,104 84,90 99,350 100,519 101,672 106,901 109,953 187,1015 191,960 214,758 226,1006 238,513 255,1106 376,497 388,498 474,958 495,549 496,550 573,604 1030,1100
X(36) = midpoint of X(1) and X(484)
X(36) = reflection of X(I) in X(J) for these (I,J): (1,1319), (484,1155)
X(36) = isogonal conjugate of X(80)
X(36) = inverse-in-circumcircle of X(1)
X(36) = inverse-in-incircle of X(942)
X(36) = inverse-in-Bevan-circle of X(46)
X(36) = X(I)-Ceva conjugate of X(J) for these (I,J): (88,6), (104,1)
X(36) = crosspoint of X(58) and X(106)
X(36) = crosssum of X(I) and X(J) for these (I,J): (1,484), (10,519), (11,900)
X(36) = crossdifference of any two points on line X(37)X(650)
X(36) = X(104)-aleph conjugate of X(36)
X(36) = X(I)-beth conjugate of X(J) for these (I,J): (21,36), (100,36), (643,519)
X(37) = CROSSPOINT OF INCENTER AND CENTROID
Trilinears b + c : c + a : a + bBarycentrics a(b + c) : b(c + a) : c(a + b)
Let A'B'C' be the cevian triangle of X(1). Let A" be the centroid of triangle AB'C', and define B" and C" cyclically. Then the lines AA", BB", CC" concur in X(37). (Eric Danneels, Hyacinthos 7892, 9/13/03)
If you have The Geometer's Sketchpad, you can view X(37).
X(37) lies on these lines:
1,6 2,75 3,975 7,241 8,941 10,594 12,225 19,25 21,172 35,267 38,354 39,596 12,225 41,584 48,205 63,940 65,71 73,836 78,965 82,251 86,190 91,498 100,111 101,284 141,742 142,1086 145,391 158,281 171,846 226,440 256,694 347,948 &n