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 1998 book Triangle Centers and Central Triangles. For subsequent developments, click Links (one of the buttons atop this page). In particular, 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. For an interactive dynamic version of ETC, visit Triangle Centers with C.a.R..
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.
The algebraic definition of triangle center ( MathWorld) admits points whose geometric interpretation for fixed numerical sidelengths a,b,c is not "central." Roger Smyth offers this example: on the domain of scalene triangles, define f(a,b,c) = 1 for a>b and a>c and f(a,b,c) = 0 otherwise; then f(a,b,c) : f(b,c,a) : f(c,a,b) is a triangle center which picks out the vertex opposite the longest side. Such centers turn out to be useful, as, for example, when distinguishing between the Fermat point and the 1st isogonic center; see the note at X(13).
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 all Z-cubics (e.g., Thomson, Darboux, Napoleon, Neuberg) and 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 54,3460 60,110 61,203 62,202 64,1439 69,1245 71,579 74,346475,86 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 185,296 188,361 190,537 195,3467 196,207 201,212 204,1712 224,377 227,1465 228,1730 229,267 256,511 257,385 280,1256 281,282 282,1249 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 399,3065 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 1073,3341 1075,1148 1106,1476 1157,3483 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 2120,3461 2130,3347 3183,3345 3342,3343 3344,3351 3346,3353 3348,3472 3350,3352 3354,3355 3462,3469
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 the Thomson cubic and 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 1073,1249 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 3341,3342 3343,3344 3349,3350 3351,3352
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), (668,3227), (670,3228)
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 the Thomson cubic, the Darboux cubic, the Napoleon cubic, the Neuberg cubic, and 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 84,1490 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 616,3480 617,3479 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 1249,3344 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 2120,3463 2130,3348 2131,3355 3182,3345 3183,3346 3341,3352 3343,3349 3347,3353 3350,3356 3354,3472 3460,3469 3464,3466 3481,3484
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) = anticomplementary conjugate of X(2888)
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 the Thomson cubic, the Darboux cubic, the Napoleon cubic, the Neuberg cubic, and 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 84,3182 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 282,3351 371,485 372,486 390,495 394,1217 399,1263 477,1304 484,3483 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 1073,3350 1138,2132 1157,3482 1160,1162 1161,1163 1251,1832 1276,1277 1329,1376 1340,1348 1341,1349 1385,1538 1430,1468 1499,1550 1715,1730 1716,1721 1717,1718 1726,1782 2120,3459 2130,3346 3065,3464 3345,3353 3347,3472 3348,3355 3460,3467
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), (146), (3448), (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) = complementary conjugate of X(2883)
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 these sketches: Nine-point center, Euler Line, Roll Circle, MacBeath Inconic
X(5) lies on the Napoleon cubic (also known as the Feuerbach cubic) and 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 113,125 116,118 117,124 122,133 127,132 128,137 129,130 131,136 141,211 142,971 156,184 182,206 183,315 195,3459 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 2120,2121 3336,3467 3460,3461 3462,3463 3468,3469 3470,3471
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), (399,3448)
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), (54, 2121), (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 the Thomson cubic and 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 1073,3343 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 3342,3351 3344,3350
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) = complementary conjugate of X(2884)
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) = complementary conjugate of X(2885)
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 the Thomson cubic and 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 3341,3344 3343,3352 3349,3351
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), (6,3416), (10,3421), (55,3419), (65,72), (80,100), (2948,3448)
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) = anticomplementary conjugate of X(2891)
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), (9,3254), (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 < 2π/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, however, A> 2π/3, then the Fermat point, defined geometrically as the minimizer of |AX| + |BX| + |CX|, is not the 1st isogonic center (which is defined by the above trilinears). Trilinears for the Fermat point when A> 2π/3 are simply 1:0:0. To represent the Fermat point in the form f(a,b,c) : f(b,c,a) : f(c,a,b), one must use Boolean variables, as shown at Fermat point.
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 the Neuberg cubic and these lines:
2,16 3,17 4,61 5,18 6,14 11,202 15,30 16,616 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 the Neuberg cubic and these lines:
2,15 3,18 4,62 5,17 6,13 11,203 15,617 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 the Neuberg cubic and these lines:
1,1251 2,14 3,6 4,17 13,30 14,617 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 the Neuberg cubic and these lines:
1,1250 2,13 3,6 4,18 13,616 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 the Napoleon cubic and these lines:
2,62 3,13 4,15 5,14 6,18 12,203 16,140 62,627 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 the Napoleon cubic and these lines:
2,61 3,14 4,16 5,13 6,17 12,202 15,140 61,628 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
= 2 cos A - sin B sin C : 2 cos B - sin C sin A : 2 cos C - sin A sin 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 the Darboux cubic and 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 1490,3345 1498,3346 1610,1633 2130,2131 3182,3347 3183,3348 3353,3354 3472,3473
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) = anticomplementary conjugate of X(2892)
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) = inverse-in-orthocentric circle of X(1594)
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-circumcircle of X(2073)
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) = inverse-in-circumcircle of X(2074)
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) = inverse-in-circumcircle of X(2075)
X(29) = complement of X(3152)
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= 3 cos A - 2 sin B sin C : 3 cos B - 2 sin C sin A : 3 cos C - 2 sin A sin 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 the Neuberg cubic and 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 399,1138 484,3065 485,1151 486,1152 489,638 490,637 497,999 511,512 551,946 553,942 582,1724 599,1350 616,3440 617,3441 618,623 619,624 620,625 841,1302 935,1297 944,962 1043,1330 1141,1157 1155,1737 1157,1263 1294,1304 1319,1387 1337,3479 1338,3480 1351,1353 1465,1877 1838,1852 2132,2133 3465,3466 3481,3482
X(30) = orthopoint of X(523)
X(30) = isogonal conjugate of X(74)
X(30) = isotomic conjugate of X(1494)
X(30) = anticomplement of X(2887)
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 145,391 158,281 171,846 226,440 256,694 347,948 513,876 517,573 537,551 579,942 626,746 665,900 971,991
X(37) is the {X(1),X(9)}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(37), click More at the top of this page.
X(37) = midpoint of X(I) and X(J) for these (I,J): (75,192), (190,335)
X(37) = isogonal conjugate of X(81)
X(37) = isotomic conjugate of X(274)
X(37) = complement of X(75)
X(37) = complementary conjugate of X(2887)
X(37) = X(I)-Ceva conjugate of X(J) for these (I,J):
(1,42), (2,10), (4,209), (9,71), (10,210), (190,513), (226,65), (321,72), (335,518)
X(37) = cevapoint of X(213) and X(228)
X(37) = X(I)-cross conjugate of X(J) for these (I,J): (42,65), (228,72)
X(37) = crosspoint of X(I) and X(J) for these (I,J): (1,2), (9,281), (10,226)
X(37) = X(1)-line conjugate of X(238)
X(37) = crosssum of X(I) and X(J) for these (I,J): (1,6), (57,222), (58,284), (1333,1437)
X(37) = crossdifference of any two points on line X(36)X(238)
X(37) = X(10)-Hirst inverse of X(740)
X(37) = X(1)-aleph conjugate of X(1051)
X(37) = X(I)-beth conjugate of X(J) for these (I,J): (9,37), (644,37), (645,894), (646,37), (1018,37)
X(38) = CROSSPOINT OF X(1) AND X(75)
Trilinears b2 + c2 : c2 + a2 : a2 + b2=csc A sin(A + ω) : csc B sin(B + ω) : csc C sin(C + ω)
Barycentrics a(b2 + c2) : b(c2 + a2) : c(a2 + b2)
= sin(A + ω) : sin(B + ω) : sin(C + ω)
X(38) lies on these lines:
1,21 2,244 3,976 8,986 9,614 10,596 37,354 42,518 56,201 57,612 75,310 78,988 92,240 99,745 210,899 321,726 869,980 912,1064 1038,1106
X(38) is the {X(1),X(63)}-harmonic conjugate of X(31). For a list of other harmonic conjugates of X(38), click More at the top of this page.
X(38) = isogonal conjugate of X(82)
X(38) = isotomic conjugate of X(3112)
X(38) = anticomplement of X(1215)
X(38) = crosspoint of X(1) and X(75)
X(38) = crosssum of X(1) and X(31)
X(38) = crossdifference of any two points on line X(661)X(830)
X(38) = X(643)-beth conjugate of X(38)
X(39) = BROCARD MIDPOINT
Trilinears a(b2 + c2) : b(c2 + a2) : c(a2 + b2)= 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)
Barycentrics a2(b2 + c2) : b2(c2 + a2) : c2(a2 + b2)
The midpoint of the 1st and 2nd Brocard points, given by trilinears c/b : a/c : b/a and b/c : c/a : a/b. The third and fourth trilinear representations were given by Peter J. C. Moses (10/3/03); cf. X(511), X(32), X(182).
X(39) lies on these lines:
1,291 2,76 3,6 4,232 5,114 9,978 10,730 36,172 37,596 51,237 54,248 83,99 110,755 140,230 141,732 185,217 213,672 325,626 395,618 493,494 512,881 588,589 590,642 597,1084 615,641
X(39) is the {X(3),X(6)}-harmonic conjugate of X(32). For a list of other harmonic conjugates of X(39), click More at the top of this page.
Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 10: The Brocard Points.
X(39) = midpoint of X(76) and X(194)
X(39) = isogonal conjugate of X(83)
X(39) = isotomic conjugate of X(308)
X(39) = inverse-in-circumcircle of X(2076)
X(39) = inverse-in-Brocard-circle of X(32)
X(39) = inverse-in-1st-Lemoine-circle of X(2458)
X(39) = complement of X(76)
X(39) = complementary conjugate of X(626)
X(39) = eigencenter of anticevian triangle of X(512)
X(39) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,141), (4,211), (99,512)
X(39) = crosspoint of X(I) and X(J) for these (I,J): (2,6), (141,427)
X(39) = crosssum of X(I) and X(J) for these (I,J): (2,6), (251,1176)
X(39) = crossdifference of any two points on line X(661)X(830)
X(40) = BEVAN POINT
Trilinears cos B + cos C - cos A - 1 : cos C + cos A - cos B - 1 : cos A + cos B - cos C - 1= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b/(c + a - b) + c/(a + b - c) - a/(b + c - a)
= g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = sin2(B/2) + sin2(C/2) - sin2(A/2)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
If you have The Geometer's Sketchpad, you can view X(40).
X(40) = point of concurrence of the perpendiculars from the excenters to the respective sides
X(40) = circumcenter of the excentral triangle
X(40) = incenter of the extangents triangle if triangle ABC is acute
X(40) = perspector of the excentral and extangents triangles
This point is mentioned in a problem proposal by Benjamin Bevan, published in Leybourn's Mathematical Repository, 1804, p. 18.
X(40) lies on the Darboux cubic and these lines:
1,3 2,926 4,9 6,380 8,20 30,191 31,580 33,201 34,212 42,581 43,970 58,601 64,72 77,947 78,100 80,90 92,412 101,972 108,207 109,255 164,188 190,341 196,208 219,610 220,910 221,223 256,989 376,519 386,1064 387,579 390,938 392,474 511,1045 550,952 595,602 728,1018 936,960 958,1012 978,1050 1498,3345 2130,33542131,3472 3182,3346 3183,3347 3348,3353 3355,3473
X(40) is the {X(55),X(65)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(40), click More at the top of this page.
X(40) = midpoint of X(8) and X(20)
X(40) = reflection of X(I) in X(J) for these (I,J): (1,3), (4,10), (84,1158), (962,946), (1482,1385)
X(40) = isogonal conjugate of X(84)
X(40) = isotomic conjugate of X(309)
X(40) = inverse-in-circumcircle of X(2077)
X(40) = complement of X(962)
X(40) = anticomplement of X(946)
X(40) = X(I)-Ceva conjugate of X(J) for these (I,J): (8,1), (20,1490), (63,9), (347,223)
X(40) = X(I)-cross conjugate of X(J) for these (I,J): (198,223), (221,1)
X(40) = crosspoint of X(I) and X(J) for these (I,J): (329,347)
X(40) = crosssum of X(56) and X(1413)
X(40) = crossdifference of any two points on line X(650)X(1459)
X(40) = X(I)-aleph conjugate of X(J) for these (I,J): (1,978), (2,57), (8,40), (188,1), (556,63)
X(40) = X(I)-beth conjugate of X(J) for these (I,J): (8,4), (40,221), (643,78), (644,728)
X(41) = X(6)-CEVA CONJUGATE OF X(31)
Trilinears a2(b + c - a) : b2(c + a - b) : c2(a + b - c)= a2cot(A/2) : b2cot(B/2) : c2cot(C/2)
Barycentrics a3(b + c - a) : b3(c + a - b) : c3(a + b - c)
If you have The Geometer's Sketchpad, you can view X(41).
X(41) lies on these lines: 1,101 3,218 6,48 9,21 25,42 31,32 37,584 55,220 58,609 65,910 219,1036 226,379 560,872 601,906 603,911 663,884
X(41) is the {X(32),X(213)}-harmonic conjugate of X(31). For a list of other harmonic conjugates of X(41), click More at the top of this page.
X(41) = isogonal conjugate of X(85)
X(41) = X(I)-Ceva conjugate of X(J) for these (I,J): (6,31), (9,212), (284,55)
X(41) = crosspoint of X(I) and X(J) for these (I,J): (6,55), (9,33)
X(41) = crosssum of X(I) and X(J) for these (I,J): (1,169), (2,7), (57,77), (92,342), (226,1441), (514,1111)
X(41) = crossdifference of any two points on line X(522)X(693)
X(41) = X(I)-beth conjugate of X(J) for these (I,J): (41,32), (101,41), (220,220)
X(42) CROSSPOINT OF INCENTER AND SYMMEDIAN POINT
Trilinears a(b + c) : b(c + a) : c(a + b)= (1 + cos A)(cos B + cos C) : (1 + cos B)(cos C + cos A) : (1 + cos C)(cos A + cos B)
Barycentrics a2(b + c) : b2(c + a) : c2(a + b)
If you have The Geometer's Sketchpad, you can view X(42).
X(42) lies on these lines:
1,2 3,967 6,31 9,941 25,41 33,393 35,58 37,210 38,518 40,581 48,197 57,1001 65,73 81,100 101,111 165,991 172,199 181,228 244,354 308,313 321,740 517,1064 560,584 649,788 694,893 748,1001 750,940 894,1045 942,1066
X(42) is the {X(1),X(43)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(42), click More at the top of this page.
X(42) = reflection of X(321) in X(1215)
X(42) = isogonal conjugate of X(86)
X(42) = isotomic conjugate of X(310)
X(42) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,37), (6,213), (10,71), (55,228)
X(42) = crosspoint of X(I) and X(J) for these (I,J): (1,6), (33,55), (37,65)
X(42) = crosssum of X(I) and X(J) for these (I,J): (1,2), (7,77), (21,81)
X(42) = crossdifference of any two points on line X(239)X(514)
X(42) = X(1)-line conjugate of X(239)
X(42) = X(I)-beth conjugate of X(J) for these (I,J): (21,551), (55,42), (100,226), (210,210), (643,171)
X(43) = X(6)-CEVA CONJUGATE OF X(1)
Trilinears ab + ac - bc : bc + ba - ca : ca + cb - ab= csc B + csc C - csc A : csc C + csc A - csc B : csc A + csc B - csc C
Barycentrics a(ab + ac - bc) : b(bc + ba - ca) : c(ca + cb - ab)
X(43) lies on these lines:
1,2 6,87 9,256 31,100 35,1011 40,970 46,851 55,238 57,181 58,979 72,986 75,872 81,750 165,573 170,218 210,984 312,740 518,982
X(43) is the {X(2),X(42)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(43), click More at the top of this page. X(43) is the external center of similitude of the Bevan circle and Apollonius circle; the internal center is X(1695).
X(43) = reflection of X(1) in X(995)
X(43) = isogonal conjugate of X(87)
X(43) = X(6)-Ceva conjugate of X(1)
X(43) = X(192)-cross conjugate of X(1)
X(43) = crosssum of X(2) and X(330)
X(43) = X(55)-Hirst inverse of X(238)
X(43) = X(I)-aleph conjugate of X(J) for these (I,J):
(1,9), (6,43), (55,170), (100,1018), (174,169), (259,165), (365,1), (366,63), (507,362), (509,57)
X(43) = X(660)-beth conjugate of X(43)
X(44) = X(6)-LINE CONJUGATE OF X(1)
Trilinears b + c - 2a : c + a - 2b : a + b - 2cBarycentrics a(b + c - 2a) : b(c + a - 2b) : c(a + b - 2c)
X(44) lies on these lines: 1,6 2,89 10,752 31,210 51,209 65,374 88,679 181,375 190,239 193,344 214,1017 241,651 292,660 354,748 513,649 527,1086 583,992 678,902
X(44) is the {X(1),X(9)}-harmonic conjugate of X(45). For a list of other harmonic conjugates of X(44), click More at the top of this page.
X(44) = midpoint of X(I) and X(J) for these (I,J): (190,239), (3218,3257)
X(44) = reflection of X(1279) in X(238)
X(44) = isogonal conjugate of X(88)
X(44) = complement of X(320)
X(44) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,214), (88,1), (104,55)
X(44) = crosspoint of X(I) and X(J) for these (I,J): (1,88), (2,80)
X(44) = crosssum of X(I) and X(J) for these (I,J): (1,44), (6,36), (57,1465)
X(44) = crossdifference of any two points on line X(1)X(513)
X(44) = X(6)-line conjugate of X(1)
X(44) = X(88)-cross conjugate of X(44)
X(44) = X(I)-beth conjugate of X(J) for these (I,J): (9,44), (644,44), (645,239), (44,44)
X(45) = X(9)-BETH CONJUGATE OF X(1)
Trilinears 2b + 2c - a : 2c + 2a - b : 2a + 2b - cBarycentrics a(2b + 2c - a) : b(2c + 2a - b) : c(2a + 2b - c)
X(45) lies on these lines: 1,6 2,88 53,281 55,678 141,344 198,1030 210,968 346,594
X(45) is the {X(1),X(9)}-harmonic conjugate of X(44). For a list of other harmonic conjugates of X(45), click More at the top of this page.
X(45) = isogonal conjugate of X(89)
X(45) = crosssum of X(6) and X(999)
X(45) = X(I)-beth conjugate of X(J) for these (I,J): (9,1), (644,45)
X(46) X(4)-CEVA CONJUGATE OF X(1)
Trilinears cos B + cos C - cos A : cos C + cos A - cos B : cos A + cos B - cos CBarycentrics a(cos B + cos C - cos A) : b(cos C + cos A - cos B) : c(cos A + cos B - cos C)
X(46) lies on these lines:
1,3 4,90 9,79 10,63 19,579 34,47 43,851 58,998 78,758 80,84 100,224 158,412 169,672 200,1004 218,910 222,227 225,254 226,498 269,1103 404,997 474,960 499,946 595,614 750,975 978,1054
X(46) is the {X(3),X(65)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(46), click More at the top of this page.
X(46) = reflection of X(I) in X(J) for these (I,J): (1,56), (1479,1210)
X(46) = isogonal conjugate of X(90)
X(46) = inverse-in-Bevan-circle of X(36)
X(46) = X(4)-Ceva conjugate of X(1)
X(46) = crosssum of X(3) and X(1069)
X(46) = X(I)-aleph conjugate of X(J) for these (I,J): (4,46), (174,223), (188,1079), (366,610), (653, 1020)
X(46) = X(100)-beth conjugate of X(46)
X(47) = X(110)-BETH CONJUGATE OF X(34)
Trilinears cos 2A : cos 2B : cos 2C = f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = a2(a4 + b4 + c4 - 2a2b2 - 2a2c2)
Barycentrics a cos 2A : b cos 2B : c cos 2C
X(47) lies on these lines:
1,21 19,921 33,90 34,46 35,212 36,602 91,92 158,162 171,498 238,499
X(47) is the {X(91),X(92)}-harmonic conjugate of X(564). For a list of other harmonic conjugates of X(47), click More at the top of this page.
X(47) = isogonal conjugate of X(91)
X(47) = eigencenter of cevian triangle of X(92)
X(47) = eigencenter of anticevian triangle of X(48)
X(47) = X(92)-Ceva conjugate of X(48)
X(47) = crosssum of X(I) and X(J) for these (I,J): (656,1109)
X(47) = X(275)-aleph conjugate of X(92)
X(47) = X(I)-beth conjugate of X(J) for these (I,J): (110,34), (643,47)
X(47) = trilinear product of X(371) and X(372)
X(48) = CROSSPOINT OF X(1) AND X(63)
Trilinears sin 2A : sin 2B : sin 2C= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = tan B + tan C
= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 + c2 - a2)
Barycentrics a sin 2A : b sin 2B : c sin 2C
X(48) lies on these lines:
1,19 3,71 6,41 9,101 31,560 36,579 37,205 42,197 55,154 63,326 75,336 163,1094 184,212 220,963 255,563 281,944 282,947 354,584 577,603 692,911 949,1037 958,965
X(48) is the {X(41),X(604)}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(48), click More at the top of this page.
X(48) = isogonal conjugate of X(92)
X(48) = isotomic conjugate of X(1969)
X(48) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,31), (3,212), (63,255), (92,47), (284, 6)
X(48) = X(228)-cross conjugate of X(3)
X(48) = crosspoint of X(I) and X(J) for these (I,J): (1,63), (3,222), (91,92), (219,268)
X(48) = crosssum of X(I) and X(J) for these (I,J): (1,19), (4,281), (47,48), (196,278), (523, 1146), (661,1109)
X(48) = crossdifference of any two points on line X(240)X(522)
X(48) = X(1)-line conjugate of X(240)
X(48) = X(I)-beth conjugate of X(J) for these (I,J): (101,48), (219,219), (284,604), (906,48)
X(49) = CENTER OF SINE-TRIPLE-ANGLE CIRCLE
Trilinears cos 3A : cos 3B : cos 3CBarycentrics sin A cos 3A : sin B cos 3B : sin C cos 3C
V. Thebault, "Sine-triple-angle-circle," Mathesis 65 (1956) 282-284.
X(49) lies on these lines: 1,215 3,155 4,156 5,54 24,568 52,195 93,94 381,578
X(49) is the {X(54),X(110)}-harmonic conjugate of X(5). For a list of other harmonic conjugates of X(49), click More at the top of this page.
X(49) = isogonal conjugate of X(93)
X(49) = eigencenter of cevian triangle of X(94)
X(49) = eigencenter of anticevian triangle of X(50)
X(49) = X(94)-Ceva conjugate of X(50)
X(50) = X(74)-CEVA CONJUGATE OF X(184)
Trilinears sin 3A : sin 3B : sin 3CBarycentrics sin A sin 3A : sin B sin 3B : sin C sin 3C
X(50) lies on these lines: 3,6 67,248 112,477 115,231 230,858 338,401 647,654
X(50) is the {X(3),X(6)}-harmonic conjugate of X(566). For a list of other harmonic conjugates of X(40), click More at the top of this page.
X(50) = isogonal conjugate of X(94)
X(50) = inverse-in-Brocard-circle of X(566)
X(50) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,215), (74,184), (94,49)
X(50) = crosspoint of X(I) and X(J) for these (I,J): (93,94), (186,323)
X(50) = crosssum of X(49) and X(50)
X(50) = crossdifference of any two points on line X(5)X(523)
X(50) = barycentric product of X(15) and X(16)
X(51) = CENTROID OF ORTHIC TRIANGLE
Trilinears a2cos(B - C) : b2cos(C - A) : c2cos(A - B)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a2(b2 + c2) - (b2 - c2)2]
Barycentrics a3cos(B - C) : b3cos(C - A) : c3cos(A - B)
X(51) lies on these lines:
2,262 4,185 5,52 6,25 21,970 22,182 23,575 24,578 26,569 31,181 39,237 44,209 54,288 107,275 125,132 129,137 130,138 199,572 210,374 216,418 381,568 397,462 398,463 573,1011
X(51) is the {X(5),X(143)}-harmonic conjugate of X(52). For a list of other harmonic conjugates of X(51), click More at the top of this page.
X(51) = reflection of X(210) in X(375)
X(51) = isogonal conjugate of X(95)
X(51) = complement of X(2979)
X(51) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,53), (5,216), (6,217)
X(51) = X(217)-cross conjugate of X(216)
X(51) = crosspoint of X(I) and X(J) for these (I,J): (4,6), (5,53)
X(51) = crosssum of X(I) and X(J) for these (I,J): (2,3), (6,160), (54,97)
X(51) = crossdifference of any two points on line X(323)X(401)
X(52) = ORTHOCENTER OF ORTHIC TRIANGLE
Trilinears cos 2A cos(B - C) : cos 2B cos(C - A) : cos 2C cos(A - B)= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A (sec 2B + sec 2C)
Barycentrics tan A (sec 2B + sec 2C) : tan B (sec 2C + sec 2A) : tan C (sec 2A + sec 2B)
X(52) lies on these lines:
3,6 4,68 5,51 25,155 26,184 30,185 49,195 113,135 114,211 128,134 129,139
X(52) is the {X(5),X(143)}-harmonic conjugate of X(51). For a list of other harmonic conjugates of X(52), click More at the top of this page.
X(52) = reflection of X(I) in X(J) for these (I,J): (3,389), (5,143), (113,1112), (1209,973)
X(52) = isogonal conjugate of X(96)
X(52) = anticomplement of X(1216)
X(52) = inverse-in-Brocard-circle of X(569)
X(52) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,5), (317,467), (324,216)
X(52) = crosspoint of X(4) and X(24)
X(52) = crosssum of X(3) and X(68)
X(53) = SYMMEDIAN POINT OF ORTHIC TRIANGLE
Trilinears tan A cos(B - C) : tan B cos(C - A) : tan C cos(A - B)Barycentrics a tan A cos(B - C) : b tan B cos(C - A) : c tan C cos(A - B)
X(53) lies on these lines:
4,6 5,216 25,157 30,577 45,281 115,133 128,139 137,138 141,264 232,427 273,1086 275,288 311,324 317,524 318,594 395,472 396,473
X(53) is the {X(4),X(393)}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(53), click More at the top of this page.
X(53) = isogonal conjugate of X(97)
X(53) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,51), (324,5)
X(53) = X(51)-cross conjugate of X(5)
X(53) = crosssum of X(3) and X(577)
X(54) = KOSNITA POINT
Trilinears sec(B - C) : sec(C - A) : sec(A - B)Barycentrics sin A sec(B - C) : sin B sec(C - A) : sin C sec(A - B)
John Rigby, "Brief notes on some forgotten geometrical theorems," Mathematics and Informatics Quarterly 7 (1997) 156-158.
Let O be the circumcenter of triangle ABC, and Oa the circumcenter of triangle BOC. Define Ob and Oc cyclically. Then the lines AOa, BOb, COc concur in X(54). For details and generalization, see
Darij Grinberg, A New Circumcenter Question
X(54) lies on the Napoleon cubic and these lines:
1,3460 2,68 3,97 4,184 5,49 6,24 12,215 36,73 39,248 51,288 64,378 69,95 71,572 72,1006 74,185 112,217 140,252 156,381 186,389 276,290 575,895 627,628 826,879 3336,3468
X(54) is the {X(5),X(49)}-harmonic conjugate of X(110). For a list of other harmonic conjugates of X(54), click More at the top of this page.
X(54) = midpoint of X(3) and X(195)
X(54) = reflection of X(195) in X(1493)
X(54) = isogonal conjugate of X(5)
X(54) = isotomic conjugate of X(311)
X(54) = inverse-in-circumcircle of X(1157)
X(54) = complement of X(2888)
X(54) = anticomplement of X(1209)
X(54) = X(I)-Ceva conjugate of X(J) for these (I,J): (5,2120), (95,97), (288,6)
X(54) = cevapoint of X(6) and X(184)
X(54) = X(I)-cross conjugate of X(J) for these (I,J): (3,96), (6,275), (186,74), (389,4), (523,110)
X(54) = crosspoint of X(95) and X(275)
X(54) = crosssum of X(I) and X(J) for these (I,J): (3,195), (51,216), (627,628)
X(55) = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)
Trilinears a(b + c - a) : b(c + a - b) : c(a + b - c)= 1 + cos A : 1 + cos B : 1 + cos C
= cos2(A/2) : cos2(B/2) : cos2(B/2)
= tan(B/2) + tan(C/2) : tan(C/2) + tan(A/2) : tan(A/2) + tan(B/2)
Barycentrics a2(b + c - a) : b2(c + a - b) : c2(a + b - c)
X(55) = center of homothety of three triangles: tangential, intangents, and extangents. Also, X(55) is the pole-with-respect-to-the-circumcircle of the trilinear polar of X(1). These properties and others are given in
O. Bottema and J. T. Groenman, "De gemene raaklijnen van de vier raakcirkels van een driehoek, twee aan twee," Nieuw Tijdschrift voor Wiskunde 67 (1979-80) 177-182.
X(55) lies on these lines:
1,3 2,11 4,12 5,498 6,31 8,21 9,200 10,405 15,203 16,202 19,25 20,388 30,495 34,227 41,220 43,238 45,678 48,154 63,518 64,73 77,1037 78,960 81,1002 92,243 103,109 104,1000 108,196 140,496 181,573 182,613 183,350 184,215 192,385 199,1030 201,774 204,1033 219,284 226,516 255,601 256,983 329,1005 376,1056 386,595 392,997 411,962 511,611 515,1012 519,956 574,1015 603,963 631,1058 650,884 654,926 748,899 840,901 846,984 869,893 1026,1083 1070,1076 1072,1074
X(55) is the {X(1),X(3)}-harmonic conjugate of X(56). For a list of other harmonic conjugates of X(55), click More at the top of this page.
X(55) = reflection of X(1478) in X(495)
X(55) = isogonal conjugate of X(7)
X(55) = inverse-in-circumcircle of X(1155)
X(55) = complement of X(3434)
X(55) = anticomplement of X(2886)
X(55) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,6), (3,198), (7,218), (8,219), (9,220), (21,9), (59,101), (104,44), (260,259), (284,41)
X(55) = cevapoint of X(42) and X(228) for these (I,J)
X(55) = X(I)-cross conjugate of X(J) for these (I,J): (41,6), (42,33), (228,212)
X(55) = crosspoint of X(I) and X(J) for these (I,J): (1,9), (3,268), (7,277), (8,281), (21,284), (59,101)
X(55) = crosssum 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), (241,1362), (513,1086), (905,1364), (1361,1465)
X(55) = crossdifference of any two points on line X(241)X(514)
X(55) = X(I)-Hirst inverse of X(J) for these (I,J): (6,672), (43,241)
X(55) = X(1)-line conjugate of X(241)
X(55) = X(I)-beth conjugate of X(J) for these (I,J): (21,999), (55,31), (100,55), (200,200), (643,2), (1021,1024)
X(56) = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)
Trilinears a/(b + c - a) : b/(c + a - b) : c/(a + b - c)= 1 - cos A : 1 - cos B : 1 - cos C
= sin2(A/2) : sin2(B/2) : sin2(C/2)
Barycentrics a2/(b + c - a) : b2/(c + a - b) : c2/(a + b - c)
The perspector of the tangential triangle and the reflection of the intangents triangle in X(1).
X(56) lies on these lines:
1,3 2,12 4,11 5,499 6,41 7,21 8,404 10,474 19,207 20,497 22,977 25,34 28,278 30,496 31,154 32,1015 33,963 38,201 58,222 61,202 62,203 63,960 72,997 77,1036 78,480 81,959 85,870 87,238 100,145 101,218 105,279 106,109 140,495 181,386 182,611 197,227 212,939 219,579 220,672 223,937 226,405 255,602 266,289 269,738 330,385 376,1058 411,938 511,613 551,553 607,911 631,1056 667,764 946,1012 978,979 1025,1083 1070,1074 1072,1076
X(56) is the {X(1),X(3)}-harmonic conjugate of X(55). For a list of other harmonic conjugates of X(56), click More at the top of this page.
X(56) = midpoint of X(1) and X(46)
X(56) = reflection of X(I) in X(J) for these (I,J): (1479,496)
X(56) = isogonal conjugate of X(8)
X(56) = inverse-in-circumcircle of X(1319)
X(56) = complement of X(3436)
X(56) = anticomplement of X(1329)
X(56) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,221), (7,222), (28,34), (57,6), (59,109), (108,513)
X(56) = X(31)-cross conjugate of X(6)
X(56) = crosspoint of X(I) and X(J) for these (I,J): (1,84), (7,278), (28,58), (57,269), (59,109)
X(56) = crosssum 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), (519,1145)
X(56) = crossdifference of any two points on line X(522)X(650)
X(56) = X(I)-Hirst inverse of X(J) for these (I,J): (6,1458), (34,1430), (57,1429), (604,1428), (1416,1438)
X(56) = X(266)-aleph conjugate of X(1050)
X(56) = X(I)-beth conjugate of X(J) for these (I,J):
(1,1), (21,3), (56,1106), (58,56), and (P,57) for all P on the circumcircle
X(57) ISOGONAL CONJUGATE OF X(9)
Trilinears 1/(b + c - a) : 1/(c + a - b) : 1/(a + b - c)= tan(A/2) : tan(B/2) : tan(C/2)
Barycentrics a/(b + c - a) : b/(c + a - b) : c/(a + b - c)
X(57) is the perspector of the intouch triangle and excentral triangle.
X(57) lies on the Thomson cubic and these lines:
1,3 2,7 4,84 6,222 10,388 19,196 20,938 27,273 28,34 31,105 33,103 38,612 42,1002 43,181 72,474 73,386 77,81 78,404 79,90 85,274 88,651 92,653 164,177 169,277 173,174 200,518 201,975 234,362 239,330 255,580 279,479 282,3343 345,728 497,516 499,920 649,1024 658,673 748,896 758,997 955,991 957,995 959,1042 961,1106 978,1046 1020,1086 1073,3351 3342,3350
X(57) is the {X(2),X(7)}-harmonic conjugate of X(226). For a list of other harmonic conjugates of X(57), click More at the top of this page.
X(57) = midpoint of X(497) and X(3474)
X(57) = reflection of X(I) in X(J) for these (I,J): (1,999), (200,1376)
X(57) = isogonal conjugate of X(9)
X(57) = isotomic conjugate of X(312)
X(57) = inverse-in-circumcircle of X(2078)
X(57) = inverse-in-Bevan-circle of X(1155)
X(57) = complement of X(329)
X(57) = anticomplement of X(3452)
X(57) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,223), (7,1), (27,278), (81,222), (85,77), (273,34), (279,269)
X(57) = cevapoint of X(I) and X(J) for these (I,J): (6,56), (19,208)
X(57) = X(I)-cross conjugate of X(J) for these (I,J): (6,1), (19,84), (56,269), (65,7)
X(57) = crosspoint of X(I) and X(J) for these (I,J): (2,189), (7,279), (27,81), (85,273)
X(57) = crosssum of X(I) and X(J) for these (I,J): (1,165), (6,198), (31,205), (37,71), (41,212), (55,220), (210,1334)
X(57) = crossdifference of any two points on line X(650)X(663)
X(57) = X(I)-Hirst inverse of X(J) for these (I,J): (1,241), (7,1447), (56,1429), (105,1462), (910,1419)
X(57) = X(I)-aleph conjugate of X(J) for these (I,J):
(2,40), (7,57), (57,978), (174,1), (366,165), (507,503), (508,9), (509,43)
X(57) = X(I)-beth conjugate of X(J) for these (I,J):
(2,2), (81,57), (88,57), (100,57), (110,31), (162,57), (190,57), (333,63), (648,92), (651,57), (653,57), (655,57), (658,57), (660,57), (662,57), (673,57), (771,57), (799,57), (823,57), (897, 57)
X(58) ISOGONAL CONJUGATE OF X(10)
Trilinears a/(b + c) : b/(c + a) : c/(a + b)Barycentrics a2/(b + c) : b2/(c + a) : c2/(a + b)
X(58) is the point of concurrence of the Brocard axes of triangles BIC, CIA, AIB, ABC, (where I denotes the incenter, X(1)), as proved in
Antreas P. Hatzipolakis, Floor van Lamoen, Barry Wolk, and Paul Yiu, Concurrency of Four Euler Lines, Forum Geometricorum 1 (2001) 59-68.
X(58) lies on these lines:
1,21 2,540 3,6 7,272 8,996 9,975 10,171 20,387 25,967 27,270 28,34 29,162 35,42 36,60 40,601 41,609 43,979 46,998 56,222 65,109 82,596 84,990 86,238 87,978 99,727 101,172 103,112 106,110 229,244 269,1014 274,870 314,987 405,940 519,1043 942,1104 977,982 1019,1027
X(58) is the {X(3),X(6)}-harmonic conjugate of X(386). For a list of other harmonic conjugates of X(58), click More at the top of this page.
X(58) = isogonal conjugate of X(10)
X(58) = isotomic conjugate of X(313)
X(58) = inverse-in-circumcircle of X(1326)
X(58) = inverse-in-Brocard-circle of X(386)
X(58) = complement of X(1330)
X(58) = anticomplement of X(3454)
X(58) = X(I)-Ceva conjugate of X(J) for these (I,J): (81,284), (267,501), (270,28)
X(58) = cevapoint of X(6) and X(31)
X(58) = X(I)-cross conjugate of X(J) for these (I,J): (6,81), (36,106), (56,28), (513,109)
X(58) = crosspoint of X(I) and X(J) for these (I,J): (1,267), (21,285), (27,86), (60,270)
X(58) = crosssum of X(I) and X(J) for these (I,J): (1,191), (6,199), (12,201), (37,210), (42,71), (65,227), (594, 756)
X(58) = crossdifference of any two points on line X(523)X(661)
X(58) = X(6)-Hirst inverse of X(1326)
X(58) = X(I)-beth conjugate of X(J) for these (I,J): (21,21), (60,58), (110,58), (162,58), (643,58), (1098,283)
X(59) ISOGONAL CONJUGATE OF X(11)
Trilinears 1/[1 - cos(B - C)] : 1/[1 - cos(C - A)] : 1/[1 - cos(A - B)]Barycentrics a/[1 - cos(B - C)] : b/[1 - cos(C - A)] : c/[1 - cos(A - B)]
X(59) lies on these lines: 36,1110 60,1101 100,521 101,657 109,901 513,651 518,765 523,655
X(59) = isogonal conjugate of X(11)
X(59) = cevapoint of X(I) and X(J) for these (I,J): (55,101), (56,109)
X(59) = X(I)-cross conjugate of X(J) for these (I,J): (1,110), (3,100), (55,101), (56,109), (182,1492)
X(59) = X(765)-beth conjugate of X(765)
X(60) = ISOGONAL CONJUGATE OF X(12)
Trilinears 1/[1 + cos(B - C)] : 1/[1 + cos(C - A)] : 1/[1 + cos(A - B)]Barycentrics a/[1 + cos(B - C)] : b/[1 + cos(C - A)] : c/[1 + cos(A - B)]
X(60) lies on these lines: 1,110 21,960 28,81 36,58 59,1101 86,272 283,284 404,662 757,1014
X(60) = isogonal conjugate of X(12)
X(60) = X(58)-cross conjugate of X(270)
X(60) = X(I)-beth conjugate of X(J) for these (I,J): (60,849), (1098,1098)
X(61) = ISOGONAL CONJUGATE OF X(17)
Trilinears sin(A + π/6) : sin(B + π/6) : sin(C + π/6)= cos(A - π/3) : cos(B - π/3) : cos(C - π/3)
Barycentrics sin A sin(A + π/6) : sin B sin(B + π/6) : sin C sin(C + π/6)
X(61) lies on the Napoleon cubic and these lines:
1,203 2,18 3,6 4,13 5,14 18,628 30,397 56,202 140,395 299,636 302,629 618,627
X(61) is the {X(3),X(6)}-harmonic conjugate of X(62). For a list of other harmonic conjugates of X(61), click More at the top of this page.
X(61) = reflection of X(633) in X(635)
X(61) = isogonal conjugate of X(17)
X(61) = inverse-in-Brocard-circle of X(62)
X(61) = complement of X(633)
X(61) = anticomplement of X(635)
X(61) = eigencenter of cevian triangle of X(14)
X(61) = eigencenter of anticevian triangle of X(16)
X(61) = X(14)-Ceva conjugate of X(16)
X(61) = crosspoint of X(302) and X(473)
X(62) = ISOGONAL CONJUGATE OF X(18)
Trilinears sin(A - π/6) : sin(B - π/6) : sin(C - π/6)= cos(A + π/3) : cos(B + π/3) : cos(C + π/3)
Barycentrics sin A sin(A - π/6) : sin B sin(B - π/6) : sin C sin(C - π/6)
X(62) lies the Napoleon cubic and these lines:
1,202 2,17 3,6 4,14 5,13 17,627 30,398 56,203 140,396 298,635 303,630 619,628
X(62) is the {X(3),X(6)}-harmonic conjugate of X(61). For a list of other harmonic conjugates of X(62), click More at the top of this page.
X(62) = reflection of X(634) in X(636)
X(62) = isogonal conjugate of X(18)
X(62) = inverse-in-Brocard-circle of X(61)
X(62) = complement of X(634)
X(62) = anticomplement of X(636)
X(62) = eigencenter of cevian triangle of X(13)
X(62) = eigencenter of anticevian triangle of X(15)
X(62) = X(13)-Ceva conjugate of X(15)
X(62) = crosspoint of X(303) and X(472)
X(63) = ISOGONAL CONJUGATE OF X(19)
Trilinears cot A : cot B : cot C= b2 + c2 - a2 : c2 + a2 - b2 : a2 + b2 - c2
Barycentrics cos A : cos B : cos C
X(63) lies on these lines:
1,21 2,7 3,72 8,20 10,46 19,27 33,1013 36,997 37,940 48,326 55,518 56,960 65,958 69,71 77,219 91,921 100,103 162,204 169,379 171,612 190,312 194,239 201,603 210,1004 212,1040 213,980 220,241 223,651 238,614 240,1096 244,748 304,1102 318,412 354,1001 392,999 404,936 405,942 452,938 484,535 517,956 544,1018 561,799 654,918 750,756
X(63) is the {X(9),X(57)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(63), click More at the top of this page.
X(63) = reflection of X(I) in X(J) for these (I,J): (1,993), (1478,10)
X(63) = isogonal conjugate of X(19)
X(63) = isotomic conjugate of X(92)
X(63) = anticomplement of X(226)
X(63) = anticomplementary conjugate of X(2893)
X(63) = X(I)-Ceva conjugate of X(J) for these (I,J): (7,224), (69,78), (75,1), (304,326), (333,2), (348,77)
X(63) = cevapoint of X(I) and X(J) for these (I,J): (3,219), (9,40), (48,255), (71,72)
X(63) = X(I)-cross conjugate of X(J) for these (I,J): (3,77), (9,271), (48,1), (71,3), (72,69), (219,78), (255,326)
X(63) = crosspoint of X(I) and X(J) for these (I,J): (69,348), (75,304)
X(63) = crosssum of X(25) and X(607)
X(63) = crossdifference of any two points on line X(661)X(663)
X(63) = X(I)-aleph conjugate of X(J) for these (I,J):
(2,1), (75,63), (92,920), (99,662), (174,978), (190,100), (333,411), (366,43), (514,1052), (556,40), (648,162), (664,651), (668,190), (670,799), (671,897), (903,88)
X(63) = X(I)-beth conjugate of X(J) for these (I,J):
(63,222), (190,63), (333,57), (345,345), (643,63), (645,312), (662,223)
X(64) = ISOGONAL CONJUGATE OF X(20)
Trilinears 1/(cos A - cos B cos C) : 1/(cos B - cos C cos A) : 1/(cos C - cos A cos B)Barycentrics a/(cos A - cos B cos C) : b/(cos B - cos C cos A) : c/(cos C - cos A cos B)
A construction of X(64) appears in Lemoine's 1886 paper cited at X(19).
X(64) lies on the Darboux cubic and these lines:
1,3182 3,154 4,3183 6,185 20,69 24,74 30,68 33,65 40,72 54,378 55,73 71,198 84,3353 265,382 3345,3472 3346,3355
X(64) = reflection of X(1498) in X(3)
X(64) = isogonal conjugate of X(20)
X(64) = anticomplement of X(2883)
X(64) = X(25)-cross conjugate of X(6)
X(64) = X(1)-beth conjugate of X(207)
X(65) = ORTHOCENTER OF THE INTOUCH TRIANGLE
Trilinears cos B + cos C : cos C + cos A : cos A + cos B= (b + c)/(b + c - a) : (c + a)/(c + a - b) : (a + b)/(a + b - c)
= sin(A/2) cos(B/2 - C/2) : sin(B/2) cos(C/2 - A/2) : sin(C/2) cos(A/2 - B/2)
Barycentrics a(b + c)/(b + c - a) : b(c + a)/(c + a - b) : c(a + b)/(a + b - c)
X(65) is the perspector of ABC and the extangents triangle.
X(65) lies on these lines:
1,3 2,959 4,158 6,19 7,8 10,12 11,117 29,296 31,1104 33,64 37,71 41,910 42,73 44,374 58,109 63,958 68,91 74,108 77,969 79,80 81,961 110,229 169,218 172,248 224,1004 225,407 243,412 257,894 278,387 279,1002 386,994 409,1098 474,997 497,938 516,950 519,553 604,1100 651,895 1039,1041 1061,1063
X(65) is the {X(1),X(40)}-harmonic conjugate of X(55). For a list of other harmonic conjugates of X(65), click More at the top of this page.
X(65) = reflection of X(I) in X(J) for these (I,J): (1,942), (72,10)
X(65) = isogonal conjugate of X(21)
X(65) = isotomic conjugate of X(314)
X(65) = inverse-in-incircle of X(1319)
X(65) = anticomplement of X(960)
X(65) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,73), (4,225), (7,226), (10,227), (109,513), (226,37)
X(65) = X(42)-cross conjugate of X(37)
X(65) = crosspoint of X(I) and X(J) for these (I,J): (1,4), (7,57)
X(65) = crosssum of X(I) and X(J) for these (I,J): (1,3), (9,55), (56,1394)
X(65) = crossdifference of any two points on line X(521)X(650)
X(65) = X(1284)-Hirst inverse of X(1400)
X(65) = X(I)-beth conjugate of X(J) for these (I,J):
(1,65), (8,72), (10,10), (65,1042), (80,65), (100,65), (101,213), (291,65), (668,65), (1018, 65)
X(66) = ISOGONAL CONJUGATE OF X(22)
Trilinears bc/(b4 + c4 - a4) : ca/(c4 + a4 - b4) : ab/(a4 + b4 - c4)Barycentrics 1/(b4 + c4 - a4) : 1/(c4 + a4 - b4) : 1/(a4 + b4 - c4)
X(66) lies on these lines:
2,206 3,141 6,427 68,511 73,976 193,895 248,571 290,317 879,924
X(66) = midpoint of X(2892) and X(3448)
X(66) = reflection of X(I) in X(J) for these (I,J): (159,141), (1177,125)
X(66) = isogonal conjugate of X(22)
X(66) = isotomic conjugate of X(315)
X(66) = cyclocevian conjugate of X(2998)
X(66) = anticomplement of X(206)
X(66) = cevapoint of X(125) and X(512)
X(66) = X(32)-cross conjugate of X(2)
X(66) = crosssum of X(3) and X(159)
X(67) = ISOGONAL CONJUGATE OF X(23)
Trilinears bc/(b4 + c4 - a4 - b2c2) : ca/(c4 + a4 - b4 - c2a2) : ab/(a4 + b4 - c4 - a2b2)Barycentrics 1/(b4 + c4 - a4 - b2c2) : 1/(c4 + a4 - b4 - c2a2) : 1/(a4 + b4 - c4 - a2b2)
X(67) lies on these lines:
3,542 4,338 6,125 50,248 74,935 110,141 265,511 290,340 524,858 526,879
X(67) = midpoint of X(69) and X(3448)
X(67) = reflection of X(I) in X(J) for these (I,J): (6,125), (110,141)
X(67) = isogonal conjugate of X(23)
X(67) = isotomic conjugate of X(316)
X(67) = inverse-in-circumcircle of X(3455)
X(67) = cevapoint of X(141) and X(524)
X(67) = X(187)-cross conjugate of X(2)
X(68) PRASOLOV POINT
Trilinears cos A sec 2A : cos B sec 2B : cos C sec 2CBarycentrics tan 2A : tan 2B : tan 2C
Let A'B'C' be the reflection of the orthic triangle of ABC in X(5). The lines AA', BB', CC' concur in X(68), as proved in
V. V. Prasolov, Zadachi po planimetrii, Moscow, 4th edition, 2001.
Coordinates for X(68) can be obtained easily from the Ceva ratios given his Prasolov's proof of concurrence.
X(68) lies on these lines:
2,54 3,343 4,52 5,6 11,1069 20,74 26,161 30,64 65,91 66,511 73,1060 136,254 290,315 568,973
X(68) = reflection of X(155) in X(5)
X(68) = isogonal conjugate of X(24)
X(68) = isotomic conjugate of X(317)
X(68) = anticomplement of X(1147)
X(68) = X(96)-Ceva conjugate of X(3)
X(68) = cevapoint of X(I) and X(J) for these (I,J): (6,161), (125,520)
X(68) = X(115)-cross conjugate of X(525)
X(69) = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE
Trilinears (cos A)/a2 : (cos B)/b2 : (cos C)/c2= bc(b2 + c2 - a2) : ca(c2 + a2 - b2) : ab(a2 + b2 - c2)
Barycentrics cot A : cot B : cot C
= b2 + c2 - a2 : c2 + a2 - b2 : a2 + b2 - c2
X(69) lies on these lines:
2,6 3,332 4,76 7,8 9,344 10,969 20,64 22,159 54,95 63,71 72,304 73,77 74,99 110,206 125,895 144,190 150,668 189,309 192,742 194,695 200,269 248,287 263,308 265,328 274,443 290,670 297,393 347,664 350,497 404,1014 478,651 485,639 486,640 520,879
X(69) is the {X(7),X(8)}-harmonic conjugate of X(75). For a list of other harmonic conjugates of X(69), click More at the top of this page.
X(69) = reflection of X(I) in X(J) for these (I,J): (2,599), (4,1352), (6,141), (20,1350), (193,6), (895,125), (1351,5), (1353,140)
X(69) = isogonal conjugate of X(25)
X(69) = isotomic conjugate of X(4)
X(69) = cyclocevian conjugate of X(253)
X(69) = complement of X(193)
X(69) = anticomplement of X(6)
X(69) = anticomplementary conjugate of X(2)
X(69) = X(I)-Ceva conjugate of X(J) for these (I,J): (76,2), (304,345), (314,75), (332,326)
X(69) = cevapoint 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(69) = X(I)-cross conjugate of X(J) for these (I,J):
(3,2), (63,348), (72,63), (78,345), (125,525), (306,304), (307,75), (343,76)
X(69) = crosspoint of X(I) and X(J) for these (I,J): (76,305), (314,332)
X(69) = X(2)-Hirst inverse of X(325)
X(69) = X(I)-beth conjugate of X(J) for these (I,J): (69,77), (99,347), (314,7), (332,69), (645,69), (668,69)
X(70) = ISOGONAL CONJUGATE OF X(26)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/[b2 cos 2B + c2 cos 2C - a2 cos 2A]Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/[b2 cos 2B + c2 cos 2C - a2 cos 2A]
X(70) lies on this line: 74,1288
X(70) = isogonal conjugate of X(26)
X(71) = ISOGONAL CONJUGATE OF X(27)
Trilinears (b + c) cos A : (c + a) cos B : (a + b) cos CBarycentrics (b + c) sin 2A : (c + a) sin 2B : (a + b) sin 2C
X(71) lies on these lines:
1,579 3,48 4,9 6,31 35,284 37,65 54,572 63,69 64,198 74,101 165,610 190,290 583,1100
X(71) is the {X(9),X(40)}-harmonic conjugate of X(19). For a list of other harmonic conjugates of X(71), click More at the top of this page.
X(71) = isogonal conjugate of X(27)
X(71) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,228), (9, 37), (10,42), (63,72)
X(71) = X(228)-cross conjugate of X(73)
X(71) = crosspoint of X(I) and X(J) for these (I,J): (3,63), (9,219), (10,306)
X(71) = crosssum of X(I) and X(J) for these (I,J): (1,579), (4,19), (28,1127), (57,278), (58,1474)
X(71) = crossdifference of any two points on line X(242)X(514)
X(71) = X(4)-line conjugate of X(242)
X(71) = X(I)-beth conjugate of X(J) for these (I,J): (219,71), (1018,71)
X(72) = ISOGONAL CONJUGATE OF X(28)
Trilinears (b + c) cot A : (c + a) cot B : (a + b) cot C= (b + c)(b2 + c2 - a2) : (c + a)(c2 + a2 - b2) : (a + b)(a2 + b2 - c2)
Barycentrics (b + c) cos A : (c + a) cos B : (a + b) cos C
X(72) lies on these lines:
1,6 2,942 3,63 4,8 5,908 7,443 10,12 20,144 21,943 31,976 35,191 40,64 43,986 54,1006 56,997 57,474 69,304 73,201 74,100 145,452 171,1046 185,916 190,1043 222,1038 248,293 290,668 295,337 306,440 394,1060 519,950 672,1009 894,1010 940,975 978,982
X(72) is the {X(1),X(9)}-harmonic conjugate of X(405). For a list of other harmonic conjugates of X(72), click More at the top of this page.
X(72) = reflection of X(I) in X(J) for these (I,J): (1,960), (65,10), (3555,1)
X(72) = isogonal conjugate of X(28)
X(72) = isotomic conjugate of X(286)
X(72) = inverse-in-Furhmann circle of X(3419)
X(72) = anticomplement of X(942)
X(72) = X(I)-Ceva conjugate of X(J) for these (I,J): (8,10), (63,71), (69,306), (321,37)
X(72) = X(I)-cross conjugate of X(J) for these (I,J): (201,10), (228,37)
X(72) = crosspoint of X(I) and X(J) for these (I,J): (8,78), (63,69), (306,307)
X(72) = crosssum of X(I) and X(J) for these (I,J): (19,25), (34,56)
X(72) = crossdifference of any two points on line X(513)X(1430)
X(72) = X(I)-beth conjugate of X(J) for these (I,J): (8,65), (72,73), (78,72), (100,227), (644,72)
X(73) CROSSPOINT OF INCENTER AND CIRCUMCENTER
Trilinears sec B + sec C : sec C + sec A : sec A + sec BBarycentrics (cos B + cos C) sin 2A : (cos C + cos A) sin 2B : (cos A + cos B) sin 2C
X(73) lies on these lines:
1,4 3,212 6,41 21,651 35,74 36,54 37,836 42,65 55,64 57,386 66,976 68,1060 69,77 72,201 102,947 228,408 284,951 290,336 1036,1037 1057,1059
X(73) is the {X(1064),X(1066)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(73), click More at the top of this page.
X(73) = isogonal conjugate of X(29)
X(73) = X(1)-Ceva conjugate of X(65)
X(73) = X(228)-cross conjugate of X(71)
X(73) = crosspoint of X(I) and X(J) for these (I,J): (1,3), (77,222), (226,307)
X(73) = crosssum of X(I) and X(J) for these (I,J): (1,4), (33,281)
X(73) = crossdifference of any two points on line X(243)X(522)
X(73) = X(I)-Hirst inverse of X(J) for these (I,J): (1,243), (65,851)
X(73) = X(I)-beth conjugate of X(J) for these (I,J): (1,1042), (3,73), (21,946), (72,72), (100,10), (101,73), (295,73)
X(74) ISOGONAL CONJUGATE OF EULER INFINITY POINT
Trilinears 1/(cos A - 2 cos B cos C) : 1/(cos B - 2 cos C cos A) : 1/(cos C - 2 cos A cos B)= 1/(3 cos A - 2 sin B sin C) : 1/(3 cos B - 2 sin C sin A) : 1/(3 cos C - 2 sin A sin B)
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[2a4 - (b2 - c2)2 - a2(b2 + c2)]
Barycentrics a/(cos A - 2 cos B cos C) : b/(cos B - 2 cos C cos A) : c/(cos C - 2 cos A cos B)
As the isogonal conjugate of the point in which the Euler line meets the line at infinity, X(74) lies on the circumcircle.
X(74) = circumcircle-antipode of X(110)
X(74) is the point of intersection, other than A, B, and C of the circumcircle and Jerabek hyperbola.
In Hyacinthos 8129 (10/4/03), Floor van Lamoen noted that if X(74) is denoted by J, then each of the points A,B,C,J is J of the other three, in analogy with the well known property of orthocentric systems (that is, each of the points A,B,C,H is the orthocenter of the other three).
X(74) lies on the Neuberg cubic and these lines:
1,3464 2,113 3,110 4,107 6,112 20,68 24,64 30,265 35,73 54,185 65,108 67,935 69,99 71,101 72,100 98,690 187,248 477,523 484,3465 511,691 512,842 550,930 616,617 1157,3484
X(74) = midpoint of X(20) and X(3448)
X(74) = reflection of X(I) in X(J) for these (I,J): (4,125), (110,3), (146,113), (399,1511)
X(74) = isogonal conjugate of X(30)
X(74) = isotomic conjugate of X(3260)
X(74) = complement of X(146)
X(74) = anticomplement of X(113)
X(74) = cevapoint of X(I) and X(J) for these (I,J): (15,16), (50,184)
X(74) = crosssum of X(I) and X(J) for these (I,J): (3,399), (616),617)
X(74) = X(I)-cross conjugate of X(J) for these (I,J): (186,54), (526,110)
X(75) ISOTOMIC CONJUGATE OF INCENTER
Trilinears 1/a2 : 1/b2 : 1/c2= 1/(1 - cos 2A) : 1/(1 - cos 2B) : 1/(1 - cos 2C)
Barycentrics 1/a : 1/b : 1/c
This is the center X(37) of the anticomplementary triangle.
X(75) lies on these lines:
1,86 2,37 6,239 7,8 9,190 10,76 19,27 21,272 31,82 32,746 38,310 43,872 48,336 77,664 99,261 100,675 101,767 141,334 144,391 158,240 194,1107 225,264 234,556 257,698 280,309 299,554 523,876 537,668 689,745 700,971 753,789 758,994 799,897 811,1099
X(75) is the {X(7),X(8)}-harmonic conjugate of X(69). For a list of other harmonic conjugates of X(75), click More at the top of this page.
X(75) = reflection of X(I) in X(J) for these (I,J): (192,37), (335,1086), (984,10)
X(75) = isogonal conjugate of X(31)
X(75) = isotomic conjugate of X(1)
X(75) = complement of X(192)
X(75) = anticomplement of X(37)
X(75) = anticomplementary conjugate of X(2895)
X(75) = X(I)-Ceva conjugate of X(J) for these (I,J): (76,312), (274,2), (310,76), (314,69)
X(75) = cevapoint of X(I) and X(J) for these (I,J): (1,63), (2,8), (7,347), (10,321), (244,514)
X(75) = X(I)-cross conjugate of X(J) for these (I,J):
(1,92), (2,85), (7,309), (8,312), (10,2), (38,1), (63,304), (244,514), (307,69), (321,76), (347,322), (522,190)
X(75) = crosspoint of X(I) and X(J) for these (I,J): (2,330), (274,310)
X(75) = crossdifference of any two points on line X(667)X(788)
X(75) = X(I)-Hirst inverse of X(J) for these (I,J): (2,350), (334,335)
X(75) = X(83)-aleph conjugate of X(31)
X(75) = X(I)-beth conjugate of X(J) for these (I,J):
(8,984), (75,7), (99,77), (314,75), (522,876), (645,9), (646,75), (668,75), (811,342)
X(76) = 3rd BROCARD POINT
Trilinears 1/a3 : 1/b3 : 1/c3= csc(A - ω) : csc(B - ω) : csc(C - ω)
Barycentrics 1/a2 : 1/b2 : 1/c2
X(76) lies on these lines:
1,350 2,39 3,98 4,69 5,262 6,83 8,668 10,75 13,299 14,298 17,303 18,302 31,734 32,384 85,226 95,96 100,767 115,626 141,698 275,276 297,343 321,561 335,871 338,599 485,491 486,492 524,598 689,755 693,764 761,789 826,882
X(76) is the {X(2),X(194)}-harmonic conjugate of X(39). For a list of other harmonic conjugates of X(76), click More at the top of this page.
X(76) = reflection of X(194) in X(39)
X(76) = isogonal conjugate of X(32)
X(76) = isotomic conjugate of X(6)
X(76) = complement of X(194)
X(76) = anticomplement of X(39)
X(76) = anticomplementary conjugate of X(2896)
X(76) = X(I)-Ceva conjugate of X(J) for these (I,J): (308,2), (310,75)
X(76) = cevapoint of X(I) and X(J) for these (I,J): (2,69), (6,22), (75,312), (311,343), (313,321), (339,525)
X(76) = X(I)-cross conjugate of X(J) for these (I,J): (2,264), (69,305), (141,2), (321,75), (343,69), (525,99)
X(76) = crosssum of X(669) and X(1084)
X(76) = crossdifference of any two points on line X(669)X(688)
X(76) = X(I)-beth conjugate of X(J) for these (I,J): (76,85), (799,348)
X(77) = ISOGONAL CONJUGATE OF X(33)
Trilinears 1/(1 + sec A) : 1/(1 + sec B) : 1/(1 + sec C)= cos A sec2(A/2) : cos B sec2(B/2) : cos C sec2(C/2)
= (b2 + c2 - a2)/(b + c - a) : (c2 + a2 - b2)/(c + a - b) : (a2 + b2 - c2)/(a + b - c)
Barycentrics a/(1 + sec A) : b/(1 + sec B) : c/(1 + sec C)
X(77) lies on these lines:
1,7 2,189 6,241 9,651 29,34 40,947 55,1037 56,1036 57,81 63,219 65,969 69,73 75,664 102,934 283,603 309,318 738,951 988,1106 999,1057
X(77) = isogonal conjugate of X(33)
X(77) = isotomic conjugate of X(318)
X(77) = anticomplement of X(1210)
X(77) = X(I)-Ceva conjugate of X(J) for these (I,J): (85,57), (86,7), (348,63)
X(77) = cevapoint of X(I) and X(J) for these (I,J): (1,223), (3,222)
X(77) = X(I)-cross conjugate of X(J) for these (I,J): (3,63), (73,222)
X(77) = X(I)-beth conjugate of X(J) for these (I,J):
(21,990), (69,69), (86,269), (99,75), (332,326), (336,77), (662,77), (664,77), (811,77)
X(78) = ISOGONAL CONJUGATE OF X(34)
Trilinears 1/(1 - sec A) : 1/(1 - sec B) : 1/(1 - sec C)= cos A csc2(A/2) : cos B csc2(B/2) : cos C csc2(C/2)
= (b + c - a)(b2 + c2 - a2) : (c + a - b)(c2 + a2 - b2) : (a + b - c)(a2 + b2 - c2)
Barycentrics a/(1 - sec A) : b/(1 - sec B) : c/(1 - sec C)
If you have The Geometer's Sketchpad, you can view X(78).
X(78) lies on these lines:
1,2 3,63 4,908 9,21 20,329 29,33 37,965 38,988 40,100 46,758 55,960 56,480 57,404 69,73 101,205 207,653 210,958 212,283 220,949 226,377 271,394 273,322 280,282 345,1040 392,1057 474,942 517,945 644,728 999,1059
X(78) = isogonal conjugate of X(34)
X(78) = isotomic conjugate of X(273)
X(78) = X(I)-Ceva conjugate of X(J) for these (I,J): (69,63), (312,9), (332,345)
X(78) = X(I)-cross conjugate of X(J) for these (I,J): (3,271), (72,8), (212,9), (219,63)
X(78) = crosspoint of X(69) and X(345)
X(78) = crosssum of X(I) and X(J) for these (I,J): (25,608), (56,1406), (604,1395), (1042,1426)
X(78) = X(I)-beth conjugate of X(J) for these (I,J): (78,3), (643,40), (1043,1)
X(79) = ISOGONAL CONJUGATE OF X(35)
Trilinears 1/(1 + 2 cos A) : 1/(1 + 2 cos B) : 1/(1 + 2 cos C)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(b2 + c2 - a2 + bc)
= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A/2)(sin 3B/2)(sin 3C/2)
Barycentrics h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = 1/(b2 + c2 - a2 + bc)
Let A' be the reflection of X(1) in sideline BC, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(79). (Eric Danneels, Hyacinthos 7892, 9/13/03)
X(79) lies on these lines:
1,30 8,758 9,46 12,484 21,36 33,1063 34,1061 35,226 57,90 65,80 104,946 314,320 388,1000
X(79) = reflection of X(191) in X(442)
X(79) = isogonal conjugate of X(35)
X(79) = isotomic conjugate of X(319)
X(79) = cevapoint of X(481) and X(482)
X(79) = crosssum of X(55) and X(1030)
X(80) REFLECTION OF INCENTER IN FEUERBACH POINT
Trilinears 1/(1 - 2 cos A) : 1/(1 - 2 cos B) : 1/(1 - 2 cos C)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(b2 + c2 - a2 - bc)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/(b2 + c2 - a2 - bc)
X(80) lies on these lines:
1,5 2,214 7,150 8,149 9,528 10,21 30,484 33,1061 34,1063 36,104 40,90 46,84 65,79 313,314 497,1000 499,944 516,655 519,908 943,950
X(80) = midpoint of X(8) and X(149)
X(80) = reflection of X(I) in X(J) for these (I,J): (1,11), (100,10), (1317,1387)
X(80) = isogonal conjugate of X(36)
X(80) = isotomic conjugate of X(320)
X(80) = inverse-in-circumcircle of X(1387)
X(80) = inverse-in-Fuhrmann-circle of X(1)
X(80) = anticomplement of X(214)
X(80) = cevapoint of X(10) and X(519)
X(80) = X(I)-cross conjugate of X(J) for these (I,J): (44,2), (517,1)
X(80) = X(8)-beth conjugate of X(100)
X(81) CEVAPOINT OF INCENTER AND SYMMEDIAN POINT
Trilinears 1/(b + c) : 1/(c + a) : 1/(a + b)Barycentrics a/(b + c) : b/(c + a) : c/(a + b)
Let A'B'C' be the cevian triangle of X(1). Let A" be the symmedian point of triangle AB'C', and define B" and C" cyclically. Then the lines AA", BB", CC" concur in X(81). (Eric Danneels, Hyacinthos 7892, 9/13/03)
If you have The Geometer's Sketchpad, you can view X(81).
X(81) lies on these lines:
1,21 2,6 7,27 8,1010 19,969 28,60 29,189 32,980 42,100 43,750 55,1002 56,959 57,77 65,961 88,662 99,739 105,110 145,1043 226,651 239,274 314,321 377,387 386,404 411,581 593,757 715,932 859,957 941,967 982,985 1019,1022 1051,1054 1098,1104
X(81) = isogonal conjugate of X(37)
X(81) = isotomic conjugate of X(321)
X(81) = anticomplement of X(1211)
X(81) = X(I)-Ceva conjugate of X(J) for these (I,J): (7,229), (86,21), (286,28)
X(81) = cevapoint of X(I) and X(J) for these (I,J): (1,6), (57,222), (58,284)
X(81) = X(I)-cross conjugate of X(J) for these (I,J): (1,86), (3,272), (6,58), (57,27), (284,21)
X(81) = crosspoint of X(274) and X(286)
X(81) = crosssum of X(I) and X(J) for these (I,J): (1,846), (6,1030), (42,1334), (213,228)
X(81) = crossdifference of any two points on line X(512)X(661)
X(81) = X(I)-beth conjugate of X(J) for these (I,J): (333,333), (643,81), (645,81), (648,81), (662,81), (931,81)
X(82) = ISOGONAL CONJUGATE OF X(38)
Trilinears 1/(b2 + c2) : 1/(c2 + a2) : 1/(a2 + b2)= sin A csc(A + ω) : sin B csc(B + ω) : sin C csc(C + ω)
Barycentrics a/(b2 + c2) : b/(c2 + a2) : c/(a2 + b2)
X(82) lies on these lines: 1,560 10,83 31,75 37,251 58,596 689,715 759,827
X(82) = isogonal conjugate of X(38)
X(82) = isotomic conjugate of X(1930)
X(82) = cevapoint of X(1) and X(31)
X(83) CEVAPOINT OF CENTROID AND SYMMEDIAN POINT
Trilinears bc/(b2 + c2) : ca/(c2 + a2) : ab/(a2 + b2)= csc(A + ω) : csc(B + ω) : csc(C + ω)
Barycentrics 1/(b2 + c2) : 1/(c2 + a2) : 1/(a2 + b2)
X(83) lies on these lines:
2,32 3,262 4,182 5,98 6,76 10,82 17,624 18,623 39,99 213,239 217,287 275,297 597,671 689,729
X(83) = isogonal conjugate of X(39)
X(83) = isotomic conjugate of X(141)
X(83) = complement of X(2896)
X(83) = cevapoint of X(2) and X(6)
X(83) = X(I)-cross conjugate of X(J) for these (I,J): (2,308), (6,251), (512,99)
X(84) = ISOGONAL CONJUGATE OF X(40)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(cos B + cos C - cos A - 1)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
Let A',B',C' be the excenters. The perpendiculars from B' to AB and from C' to AC meet in a point A". Points B" and C" are determined cyclically. The hexyl triangle, A"B"C", is perspective to ABC, and X(84) is the perspector.
X(84) lies on the Darboux cubic and these lines: 1,221 3,9 4,57 7,946 8,20 20,40 21,285 33,603 36,90 46,80 58,990 64,3353 171,989 256,988 294,580 309,314 581,941 944,1000 2130,3345 3346,3472 3347,3355
X(84) = reflection of X(I) in X(J) for these (I,J): (40,1158), (1490,3)
X(84) = isogonal conjugate of X(40)
X(84) = isotomic conjugate of X(322)
X(84) = X(I)-Ceva conjugate of X(J) for these (I,J): (189,282), (280,1)
X(84) = X(I)-cross conjugate of X(J) for these (I,J): (19,57), (56,1)
X(84) = X(280)-aleph conjugate of X(84)
X(84) = X(I)-beth conjugate of X(J) for these (I,J): (271,3), (280,280), (285,84)
X(85) ISOTOMIC CONJUGATE OF X(9)
Trilinears b2c2/(b + c - a) : c2a2/(c + a - b) : a2b2/(a + b - c)= tan(A/2) csc2A : tan(B/2) csc2B : tan(C/2) csc2C
Barycentrics bc/(b + c - a) : ca/(c + a - b) : ab/(a + b - c)
X(85) lies on these lines:
1,664 2,241 7,8 12,120 29,34 56,870 57,274 76,226 92,331 109,767 150,355 264,309
X(85) = isogonal conjugate of X(41)
X(85) = isotomic conjugate of X(9)
X(85) = complement of X(3177)
X(85) = anticomplement of X(1212)
X(85) = X(274)-Ceva conjugate of X(348)
X(85) = cevapoint of X(I) and X(J) for these (I,J): (1,169), (2,7), (57,77), (92,342)
X(85) = X(I)-cross conjugate of X(J) for these (I,J): (2,75), (57,273), (92,309), (142,2), (226,7)
X(85) = X(I)-beth conjugate of X(J) for these (I,J):
(76,76), (85,279), (99,1), (274,85), (668,85), (789,85), (799,85), (811,85)
X(86) CEVAPOINT OF INCENTER AND CENTROID
Trilinears bc/(b + c) : ca/(c + a) : ab/(a + b)Barycentrics 1/(b + c) : 1/(c + a) : 1/(a + b)
X(86) lies on these lines:
1,75 2,6 7,21 10,319 29,34 37,190 58,238 60,272 99,106 110,675 142,284 239,1100 269,1088 283,307 310,350 741,789 870,871
X(86) = isogonal conjugate of X(42)
X(86) = isotomic conjugate of X(10)
X(86) = complement of X(1654)
X(86) = anticomplement of X(1213)
X(86) = X(274)-Ceva conjugate of X(333)
X(86) = cevapoint of X(I) and X(J) for these (I,J): (1,2), (7,77), (21,81)
X(86) = crosssum of X(1) and X(1045)
X(86) = crossdifference of any two points on line X(512)X(798)
X(86) = X(I)-cross conjugate of X(J) for these (I,J): (1,81), (2,274), (7,286), (21,333), (58,27), (513,190)
X(86) = X(I)-beth conjugate of X(J) for these (I,J): (86,1014), (99,86), (261,86), (314,314), (645,86), (811,86)
X(87) X(2)-CROSS CONJUGATE OF X(1)
Trilinears 1/(ab + ac - bc) : 1/(bc + ba - ca) : 1/(ca + cb - ab)Barycentrics a/(ab + ac - bc) : b/(bc + ba - ca) : c/(ca + cb - ab)
X(87) lies on these lines: 1,192 6,43 9,292 10,979 34,242 56,238 58,978 106,932
X(87) = isogonal conjugate of X(43)
X(87) = cevapoint of X(2) and X(330)
X(87) = X(2)-cross conjugate of X(1)
X(87) = X(932)-beth conjugate of X(87)
X(88) = ISOGONAL CONJUGATE OF X(44)
Trilinears 1/(b + c - 2a) : 1/(c + a - 2b) : 1/(a + b - 2c)Barycentrics a/(b + c - 2a) : b/(c + a - 2b) : c/(a + b - 2c)
X(88) lies on these lines: 1,100 2,45 6,89 28,162 44,679 57,651 81,662 105,901 274,799 278,653 279,658 291,660
X(88) = isogonal conjugate of X(44)
X(88) = cevapoint of X(I) and X(J) for these (I,J): (1,44), (6,36)
X(88) = X(I)-cross conjugate of X(J) for these (I,J): (44,1), (517,7)
X(88) = X(I)-aleph conjugate of X(J) for these (I,J): (88,1), (679,88), (903,63), (1022,1052)
X(88) = X(333)-beth conjugate of X(190)
X(89) = ISOGONAL CONJUGATE OF X(45)
Trilinears 1/(2b + 2c - a) : 1/(2c + 2a - b) : 1/(2a + 2b - c)Barycentrics a/(2b + 2c - a) : b/(2c + 2a - b) : c/(2a + 2b - c)
X(89) lies on these lines: 1,902 2,44 6,88 649,1022
X(89) = isogonal conjugate of X(45)
X(90) = X(3)-CROSS CONJUGATE OF X(1)
Trilinears 1/(cos B + cos C - cos A) : 1/(cos C + cos A - cos B) : 1/(cos A + cos B - cos C)Barycentrics a/(cos B + cos C - cos A) : b/(cos C + cos A - cos B) : c/(cos A + cos B - cos C)
X(90) lies on these lines: 1,155 4,46 9,35 21,224 33,47 36,84 40,80 57,79
X(90) = isogonal conjugate of X(46)
X(90) = X(3)-cross conjugate of X(1)
X(91) = ISOGONAL CONJUGATE OF X(47)
Trilinears sec 2A : sec 2B : sec 2CBarycentrics sin A sec 2A : sin B sec 2B : sin C sec 2C
X(91) lies on these lines: 19,920 31,1087 37,498 47,92 63,921 65,68 225,847 255,1109 759,925
X(91) = isogonal conjugate of X(47)
X(91) = X(48)-cross conjugate of X(92)
X(91) = trilinear product of X(485) and X(486)
X(92) CEVAPOINT OF INCENTER AND CLAWSON POINT
Trilinears csc 2A : csc 2B : csc 2CBarycentrics sec A : sec B : sec C
Let LA be the line through X(4) parallel to the internal bisector of angle A, and let
A' = BC∩LA. Define B' and C' cyclically.
Alexei Myakishev, "The M-Configuration of a Triangle," Forum Geometricorum 3 (2003) 135-144,
proves that the lines AA', BB', CC' concur in X(92). He notes that another construction follows from Proposition 2 of the article: let A1 be the midpoint of the arc BC of the circumcircle that passes through A, and let A2 be the point, other than A, in which the A-altitude meets the circumcircle. Let A" = A1A2∩BC. Define B" and C" cyclically. Then the lines AA", BB", CC" concur in X(92).
X(92) lies on these lines:
1,29 2,273 4,8 7,189 19,27 25,242 31,162 38,240 40,412 47,91 55,243 57,653 85,331 100,917 226,342 239,607 255,1087 257,297 264,306 304,561 406,1068 608,894
X(92) = isogonal conjugate of X(48)
X(92) = isotomic conjugate of X(63)
X(92) = anticomplement of X(1214)
X(92) = anticomplementary conjugate of X(2897)
X(92) = X(I)-Ceva conjugate of X(J) for these (I,J): (85, 342), (264,318), (286,4), (331,273)
X(92) = cevapoint of X(I) and X(J) for these (I,J): (1,19), (4,281), (47,48), (196,278)
X(92) = X(I)-cross conjugate of X(J) for these (I,J): (1,75), (4,273), (19,158), (48,91), (226,2), (281,318)
X(92) = crosspoint of X(I) and X(J) for these (I,J): (85,309), (264,331)
X(92) = crossdifference of every pair of points on line X(810)X(822)
X(92) = X(275)-aleph conjugate of X(47)
X(92) = X(I)-beth conjugate of X(J) for these (I,J): (92,278), (312,329), (648,57)
X(93) = ISOGONAL CONJUGATE OF X(49)
Trilinears sec 3A : sec 3B : sec 3CBarycentrics sin A sec 3A : sin B sec 3B : sin C sec 3C
X(93) lies on these lines: 4,562 49,94 186,252
X(93) = isogonal conjugate of X(49)
X(93) = X(50)-cross conjugate of X(94)
X(94) = ISOGONAL CONJUGATE OF X(50)
Trilinears csc 3A : csc 3B : csc 3CBarycentrics sin A csc 3A : sin B csc 3B : sin C csc 3C
X(94) lies on these lines: 2,300 4,143 23,98 49,93 96,925 275,324
X(94) = isogonal conjugate of X(50)
X(94) = isotomic conjugate of X(323)
X(94) = cevapoint of X(49) and X(50)
X(94) = X(I)-cross conjugate of X(J) for these (I,J): (30,264), (50,93), (265,328)
X(94) = X(300)-Hirst inverse of X(301)
X(95) CEVAPOINT OF CENTROID AND CIRCUMCENTER
Trilinears b2c2 sec(B - C) : c2a2 sec(C - A) : a2b2 sec(A - B)Barycentrics bc sec(B - C) : ca sec(C - A) : ab sec(A - B)
X(95) lies on these lines:
2,97 3,264 54,69 76,96 99,311 140,340 141,287 160,327 183,305 216,648 307,320
X(95) = isogonal conjugate of X(51)
X(95) = isotomic conjugate of X(5)
X(95) = anticomplement of X(233)
X(95) = X(276)-Ceva conjugate of X(275)
X(95) = cevapoint of X(I) and X(J) for these (I,J): (2,3), (6,160), (54,97)
X(95) = X(I)-cross conjugate of X(J) for these (I,J): (2,276), (3,97), (54,275), (140,2), (340,1494)
X(96) = ISOGONAL CONJUGATE OF X(52)
Trilinears sec 2A sec(B - C) : sec 2B sec(C - A) : sec 2C sec(A - B)Barycentrics a sec 2A sec(B - C) : b sec 2B sec(C - A) : c sec 2C sec(A - B)
X(96) lies on these lines: 2,54 4,231 24,847 76,95 94,925
X(96) = isogonal conjugate of X(52)
X(96) = cevapoint of X(3) and X(68)
X(96) = X(3)-cross conjugate of X(54)
X(97) = ISOGONAL CONJUGATE OF X(53)
Trilinears cot A sec(B - C) : cot B sec(C - A) : cot C sec(A - B)Barycentrics cos A sec(B - C) : cos B sec(C - A) : cos C sec(A - B)
X(97) lies on these lines: 2,95 3,54 110,418 216,288 276,401
X(97) = isogonal conjugate of X(53)
X(97) = isotomic conjugate of X(324)
X(97) = X(95)-Ceva conjugate of X(54)
X(97) = X(3)-cross conjugate of X(95)
Centers 74, 98- 112,
and others lie on the circumcircle. Mappings Λ and Ψ derived from such a point P for application to points X, are defined here:
Λ(P,X) = isogonal conjugate of the point where line PX meets the line at infinity.
Let Y = Λ(P,X), let Q = isogonal conjugate of P, and let Y and Z be the points where line YQ meets the circumcircle;
then Ψ(P,X) = Z.
X(98) = TARRY POINT
Trilinears sec(A + ω) : sec(B + ω) : sec(C + ω)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(b4 + c4 - a2b2 - a2c2)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/(b4 + c4 - a2b2 - a2c2)
If you have The Geometer's Sketchpad, you can view X(98).
X(98) = circumcircle-antipode of X(99)
X(98) = the point of intersection, other than A, B, and C, of the circumcircle and Kiepert hyperbola.
X(98) = Ψ(X(101), X(100)
J. W. Clawson, "Points on the circumcircle," American Mathematical Monthly 32 (1925) 169-174.
X(98) lies on these lines:
2,110 3,76 4,32 5,83 6,262 10,101 13,1080 14,383 20,148 22,925 23,94 25,107 30,671 100,228 109,171 186,935 275,427 376,543 381,598 385,511 468,685 523,842 620,631 804,878
X(98) is the {X(2),X(147)}-harmonic conjugate of X(114). For a list of harmonic conjugates, click More at the top of this page.
X(98) = midpoint between X(20) and X(148)
X(98) = reflection of X(I) in X(J) for these (I,J): (4,115), (99,3), (147,114), (1513,230)
X(98) = isogonal conjugate of X(511)
X(98) = isotomic conjugate of X(325)
X(98) = complement of X(147)
X(98) = anticomplement of X(114)
X(98) = X(290)-Ceva conjugate of X(287)
X(98) = cevapoint of X(I) and X(J) for these (I,J): (2,385), (6,237)
X(98) = X(I)-cross conjugate of X(J) for these (I,J): (230,2), (237,6), (248,287), (446,511)
X(98) = crosssum of X(385) and X(401)
X(98) = X(2)-Hirst inverse of X(287)
X(99) = STEINER POINT
Trilinears bc/(b2 - c2) : ca/(c2 - a2) : ab/(a2 - b2)= b2c2 csc(B - C) : c2a2 csc(C - A) : a2b2 csc(A - B)
Barycentrics 1/(b2 - c2) : 1/(c2 - a2) : 1/(a2 - b2)
X(99) = circumcircle-antipode of X(98)
X(99) = the point of intersection, other than A, B, and C, of the circumcircle and Steiner ellipse.
X(99) = Ψ(X(6), X(2))
If you have The Geometer's Sketchpad, you can view the following dynamic sketches:
X(99).
Steiner Circum-ellipse (showing X(99) and an area-ratio property).
For more information on the Steiner circum-ellipse, visit MathWorld.
X(99) lies on these lines:
1,741 2,111 3,76 4,114 6,729 13,303 14,302 20,147 21,105 22,305 30,316 31,715 32,194 36,350 38,745 39,83 58,727 69,74 75,261 81,739 86,106 95,311 100,668 101,190 102,332 103,1043 104,314 108,811 109,643 110,690 112,648 141,755 163,825 187,385 249,525 264,378 286,915 298,531 299,530 310,675 476,850 512,805 523,691 524,843 666,919 669,886 670,804 692,785 695,711 813,1016 889,898
X(99) is the {X(39),X(384)}-harmonic conjugate of X(83). For a list of other harmonic conjugates of X(99), click More at the top of this page.
X(99) = midpoint of X(I) and X(J) for these (I,J): (20,147), (616,617)
X(99) = reflection of X(I) in X(J) for these (I,J): (4,114), (13,619), (14,618), (98,3), (115,620), (148,115), (316,325), (385,187), (671,2)
X(99) = isogonal conjugate of X(512)
X(99) = isotomic conjugate of X(523)
X(99) = complement of X(148)
X(99) = anticomplement of X(115)
X(99) = cevapoint of X(I) and X(J) for these (I,J): (2,523), (3,525), (39,512), (100,190)
X(99) = X(1019)-cross conjugate of X(1509)
X(99) = crossdifference of any two points on line X(351)X(865)
X(99) = X(I)-cross conjugate of X(J) for these (I,J): (3,249), (22,250), (512,83), (523,2), (525,76)
X(99) = X(21)-beth conjugate of X(741)
X(100) ANTICOMPLEMENT OF FEUERBACH POINT
Trilinears 1/(b - c) : 1/(c - a) : 1/(a - b)= (a - b)(a - c) : (b - c)(b - a) : (c - a)(c - b)
Barycentrics a/(b - c) : b/(c - a) : c/(a - b)
X(100) = circumcircle-antipode of X(104)
X(100) = Ψ(X(6), X(1))
X(100) lies on these lines:
1,88 2,11 3,8 4,119 6,739 7,1004 9,1005 10,21 20,153 22,197 31,43 32,713 36,519 37,111 40,78 42,81 46,224 56,145 59,521 63,103 72,74 75,675 76,767 92,917 98,228 99,668 101,644 107,823 108,653 109,651 110,643 112,162 144,480 190,659 198,346 213,729 238,899 281,1013 329,972 442,943 484,758 513,765 516,908 517,953 518,840 522,655 560,697 594,1030 645,931 649,660 650,919 658,664 667,898 693,927 731,869 733,893 753,984 756,846 789,874 976,986
X(100) is the {X(10),X(35)}-harmonic conjugate of X(21). For a list of other harmonic conjugates of X(100), click More at the top of this page.
X(100) = midpoint of X(20) and X(153)
X(100) = reflection of X(I) in X(J) for these (I,J): (1,214), (4,119), (8,1145), (80,10), (104,3), (145,1317), (149,11), (962,1537), (1156,9), (1320,1), (1484,140)
X(100) = isogonal conjugate of X(513)
X(100) = isotomic conjugate of X(693)
X(100) = complement of X(149)
X(100) = anticomplement of X(11)
X(100) = X(99)-Ceva conjugate of X(190)
X(100) = cevapoint of X(I) and X(J) for these (I,J): (1,513), (3,521), (10,522), (142,514), (442,523)
X(100) = X(I)-cross conjugate of X(J) for these (I,J): (3,59), (513,1), (521,8), (522,21)
X(100) = crosssum of X(I) and X(J) for these (I,J): (1,1054), (244,764), (512,661), (649,663)
X(100) = crossdifference of evey pair of points on line X(244)X(665)
X(100) = X(1)-line conjugate of X(244)
X(100) = X(I)-aleph conjugate of X(J) for these (I,J):
(1,1052), (100,1), (190,63), (643,411), (666,673), (765,100), (1016,190)
X(100) = X(I)-beth conjugate of X(J) for these (I,J):
(8,80), (21,106), (100,109), (333,673), (643,100), (765,100)
X(101) = Ψ(INCENTER, SYMMEDIAN POINT)
Trilinears a/(b - c) : b/(c - a) : c/(a - b)= a(a - b)(a - c) : b(b - c)(b - a) : c(c - a)(c - b)
Barycentrics a2/(b - c) : b2/(c - a) : c2/(a - b)
X(101) = circumcircle-antipode of X(103)
X(101) = Ψ(X(1), X(6))
X(101) lies on these lines:
1,41 2,116 3,103 4,118 6,106 9,48 10,98 19,913 20,152 31,609 32,595 36,672 37,284 40,972 42,111 56,218 58,172 59,657 71,74 75,767 78,205 99,190 100,644 102,198 109,654 110,163 514,664 517,910 522,929 560,713 643,931 649,901 651,934 663,919 667,813 668,789 692,926 733,904 743,869 761,984 765,898
X(101) = midpoint of X(20) and X(152)
X(101) = reflection of X(I) in X(J) for these (I,J): (4,118), (103,3), (150,116)
X(101) = isogonal conjugate of X(514)
X(101) = isotomic conjugate of X(3261)
X(101) = complement of X(150)
X(101) = anticomplement of X(116)
X(101) = X(59)-Ceva conjugate of X(55)
X(101) = cevapoint of X(354) and X(513)
X(101) = X(I)-cross conjugate of X(J) for these (I,J): (55,59), (199,250)
X(101) = crosssum of X(I) and X(J) for these (I,J): (513,650), (523,661), (649,1459)
X(101) = crossdifference of evey pair of points on line X(11)X(244)
X(101) = X(I)-aleph conjugate of X(J) for these (I,J): (100,165), (509,1052), (662,572), (664,169)
X(101) = X(I)-beth conjugate of X(J) for these (I,J): (21,105), (644,644)
X(102) = Λ(INCENTER, ORTHOCENTER)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[sin B (sec A - sec B) + sin C (sec A - sec C)]= g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = a/[2a5 + (b + c)a4 - 2(b2 + c2)a3 - (b + c)(b2 - c2)2]
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = (sin A)/[sin B (sec A - sec B) + sin C (sec A - sec C)]
X(102) = circumcircle-antipode of X(109)
X(102) = Λ(X(1), X(4))
X(102) lies on these lines:
1,108 2,117 3,109 4,124 19,282 29,107 40,78 73,947 77,934 99,332 101,198 103,928 110,283 112,284 226,1065 516,929
X(102) = midpoint of X(20) and X(153)
X(102) = reflection of X(I) in X(J) for these (I,J): (4,124), (109,3), (151,117)
X(102) = isogonal conjugate of X(515)
X(102) = complement of X(151)
X(102) = anticomplement of X(117)
X(102) = X(21)-beth conjugate of X(108)
X(103) = ANTIPODE OF X(101)
Trilinears a/[(a - b) cot C + (a - c) cot B] : b/[(b - c) cot A + (b - a) cot C] : c/[(c - a) cot B + (c - b) cot A]= 1/(a2 - b2cos C - c2 cos B) : 1/(b2 - c2cos A - a2 cos C) : 1/(c2 - a2cos B - c2 cos A)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2/[(a - b) cot C + (a - c) cot B]
X(103) = circumcircle-antipode of X(101)
X(103) = Ψ(X(101), X(3))
X(103) lies on these lines:
1,934 2,118 3,101 4,116 20,150 27,107 33,57 55,109 58,112 63,100 99,1043 102,928 295,813 376,544 515,929 516,927 572,825 672,919 910,971
X(103) = midpoint of X(20) and X(150)
X(103) = reflection of X(I) in X(J) for these (I,J): (4,116), (101,3), (152,118)
X(103) = isogonal conjugate of X(516)
X(103) = complement of X(152)
X(103) = anticomplement of X(118)
X(103) = X(21)-beth conjugate of X(934)
X(104) = ANTIPODE OF X(100)
Trilinears 1/(-1 + cos B + cos C) : 1/(-1 + cos C + cos A) : 1/(-1 + cos C + cos B)Barycentrics a/(-1 + cos B + cos C) : b/(-1 + cos C + cos A) : c/(-1 + cos C + cos B)
X(104) = circumcircle-antipode of X(100)
X(104 is the point of intersection, other than A, B, and C, of the circumcircle and Feuerbach hyperbola
X(104) = Λ(X(1), X(3))
X(104) = Ψ(X(101), X(9)).
X(104) lies on these lines:
1,109 2,119 3,8 4,11 7,934 9,48 20,149 21,110 28,107 36,80 55,1000 79,946 99,314 105,885 112,1108 256,1064 294,919 355,404 376,528 513,953 517,901 631,958
X(104) = midpoint of X(20) and X(149)
X(104) = reflection of X(I) in X(J) for these (I,J): (4,11), (100,3), (153,119), (1537,1387)
X(104) = isogonal conjugate of X(517)
X(104) = isotomic conjugate of X(3262)
X(104) = complement of X(153)
X(104) = anticomplement of X(119)
X(104) = cevapoint of X(I) and X(J) for these (I,J): (1,36), (44,55)
X(104) = X(21)-beth conjugate of X(109)
X(105) = Λ(INCENTER, SYMMEDIAN POINT)
Trilinears 1/[b2 + c2 - a(b + c)] : 1/[c2 + a2 - b(c + a)] : 1/[a2 + b2 - c(a + b)]Barycentrics a/[b2 + c2 - a(b + c)] : b/[c2 + a2 - b(c + a)] : c/[a2 + b2 - c(a + b)]
X(105) = Λ(X(1), X(6))
X(105) = Ψ(X(101), X(1))
X(105) lies on these lines:
1,41 2,11 3,277 6,1002 21,99 25,108 28,112 31,57 56,279 81,110 88,901 104,885 106,1022 165,1054 238,291 330,932 513,840 644,1083 659,884 666,898 825,985 910,919 961,1104
X(105) = reflection of X(I) in X(J) for these (I,J): (644,1083), (1292,3)
X(105) = isogonal conjugate of X(518)
X(105) = anticomplement of X(120)
X(105) = cevapoint of X(1) and X(238)
X(105) = X(1)-Hirst inverse of X(294)
X(105) = X(I)-beth conjugate of X(J) for these (I,J): (21,101), (927,105)
X(106) = Λ(INCENTER, CENTROID)
Trilinears a/(2a - b - c) : b/(2b - c - a) : c/(2c - a - b)Barycentrics a2/(2a - b - c) : b2/(2b - c - a) : c2/(2c - a - b)
X(106) = Λ(X(1), X(2))
X(106) = Ψ(X(101), X(6))
X(106) lies on these lines:
1,88 2,121 6,101 34,108 36,901 56,109 58,110 86,99 87,932 105,1022 238,898 269,934 292,813 614,998 663,840 789,870 833,977 919,1055
X(106) = reflection of X(1293) in X(3)
X(106) = isogonal conjugate of X(519)
X(106) = isotomic conjugate of X(3264)
X(106) = anticomplement of X(121)
X(106) = X(36)-cross conjugate of X(58)
X(106) = X(I)-beth conjugate of X(J) for these (I,J): (21,100), (901,106)
X(107) = Ψ(SYMMEDIAN POINT, ORTHOCENTER)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[cos A (sin 2B - sin 2C)]= g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = bc/[(b2 - c2)(b2 + c2 - a2)2]
Barycentrics 1/[(b2 - c2)(b2 + c2 - a2)2] : 1/[(c2 - a2)(c2 + a2 - b2)2] : 1/[(a2 - b2)(a2 + b2 - c2)2]
X(107) = Ψ(X(6), X(4))
X(107) lies on these lines:
2,122 4,74 24,1093 25,98 27,103 28,104 29,102 51,275 100,823 109,162 110,648 111,393 158,759 186,477 250,687 450,511 468,842 741,1096
X(107) = reflection of X(I) in X(J) for these (I,J): (4,133), (1294,3)
X(107) = isogonal conjugate of X(520)
X(107) = isotomic conjugate of X(3265)
X(107) = anticomplement of X(122)
X(107) = cevapoint of X(4) and X(523)
X(107) = X(I)-cross conjugate of X(J) for these (I,J): (24,250), (108,162), (523,4)
X(107) = trilinear pole of line X(4)X(6)
X(108) = Ψ(CIRCUMCENTER, ORTHOCENTER)
Trilinears a/(sec B - sec C) : b/(sec C - sec A): c/(sec A - sec B)= g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = 1/[(b - c)(b + c - a)(b2 + c2 - a2)]
Barycentrics a2/(sec B - sec C) : b2/(sec C - sec A): c2/(sec A - sec B)
X(108) = Ψ(X(3), X(1))
X(108) = Ψ(X(1), X(4))
X(108) lies on these lines:
1,102 2,123 4,11 7,1013 12,451 24,915 25,105 28,225 33,57 34,106 40,207 55,196 65,74 99,811 100,653 109,1020 110,162 204,223 273,675 318,404 331,767 388,406 429,961 608,739 648,931
X(108) = reflection of X(1295) in X(3)
X(108) = isogonal conjugate of X(521)
X(108) = anticomplement of X(123)
X(108) = X(162)-Ceva conjugate of X(109)
X(108) = cevapoint of X(I) and X(J) for these (I,J): (56,513), (429,523)
X(108) = X(513)-cross conjugate of X(4)
X(108) = crosspoint of X(107) and X(162)
X(108) = crosssum of X(520) and X(656)
X(108) = X(I)-beth conjugate of X(J) for these (I,J): (21,102), (162,108)
X(109) = Ψ(INCENTER, CIRCUMCENTER)
Trilinears a/(cos B - cos C) : b/(cos C - cos A): c/(cos A - cos B)= g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = a/[(b - c)(b + c - a)]
Barycentrics a2/(cos B - cos C) : b2/(cos C - cos A): c2/(cos A - cos B)
X(109) = circumcircle-antipode of X(102)
X(109) = Λ(X(1), X(3))
X(109) = trilinear product X(1381)*X(1382)
X(109) lies on these lines:
1,104 2,124 3,102 4,117 7,675 20,151 31,57 34,46 35,73 36,953 40,255 55,103 56,106 58,65 59,901 85,767 98,171 99,643 100,651 101,654 107,162 108,1020 112,163 165,212 191,201 278,917 284,296 478,573 579,608 604,739 649,919 658,927 662,931 840,902
X(109) = midpoint of X(20) and X(151)
X(109) = reflection of X(I) in X(J) for these (I,J): (4,117), (102,3)
X(109) = isogonal conjugate of X(522)
X(109) = anticomplement of X(124)
X(109) = X(I)-Ceva conjugate of X(J) for these (I,J): (59,56), (162,108)
X(109) = cevapoint of X(65) and X(513)
X(109) = X(I)-cross conjugate of X(J) for these (I,J): (56,59), (513,58)
X(109) = crosspoint of X(110) and X(162)
X(109) = crosssum of X(I) and X(J) for these (I,J): (523,656), (652,663)
X(109) = crossdifference of evey pair of points on line X(11)X(1146)
X(109) = X(I)-aleph conjugate of X(J) for these (I,J): (100,1079), (162,580), (651,223)
X(109) = X(I)-beth conjugate of X(J) for these (I,J): (21,104), (59,109), (100,100), (110,109), (765,109), (901,109)
X(109) = trilinear product of X(1381) and X(1382)
X(110) = FOCUS OF KIEPERT PARABOLA
Trilinears csc(B - C) : csc(C - A) : csc(A - B)= a/(b2 - c2) : b/(c2 - a2) : c/(a2 - b2)
Barycentrics a2/(b2 - c2) : b2/(c2 - a2) : c2/(a2 - b2)
X(110) - circumcircle-antipode of X(74)
X(110) = isogonal conjugate of the isotomic conjugate of X(99)
X(110) = Ψ(X(6), X(3))
X(110) = Feuerbach point of the tangential triangle.
J. W. Clawson, "Points on the circumcircle," American Mathematical Monthly 32 (1925) 169-174.
Roland H. Eddy and R. Fritsch, "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," Mathematics Magazine 67 (1994) 188-205.
Benedetto Scimemi, "Paper-folding and Euler's Theorem Revisited," Forum Geometricorum.
Scimemi proves that if the Euler line is reflected in every side of triangle ABC, then the three reflections concur in X(110).
X(110) lies on these lines:
1,60 2,98 3,74 4,113 5,49 6,111 11,215 20,146 21,104 22,154 23,323 24,155 27,917 28,915 30,477 31,593 32,729 39,755 58,106 65,229 67,141 69,206 81,105 86,675 97,418 99,690 100,643 101,163 102,283 107,648 108,162 143,195 187,352 190,835 249,512 250,520 251,694 274,767 324,436 351,526 353,574 373,575 376,541 476,523 525,935 560,715 595,849 668,839 669,805 670,689 681,823 685,850 789,799 859,953
X(110) is the {X(5),X(49)}-harmonic conjugate of X(54). For a list of other harmonic conjugates of X(110), click More at the top of this page.
X(110) = midpoint of X(I) and X(J) for these (I,J): (3,399), (20,146), (23,323), (1495,3292)
X(110) = reflection of X(I) in X(J) for these (I,J): (3,1511), (4,113), (23,1495), (67,141), (74,3), (265,5), (382,1539), (895,6), (1177,206)
X(110) = isogonal conjugate of X(523)
X(110) = isotomic conjugate of X(850)
X(110) = inverse of X(2) in the Brocard circle
X(110) = complement of X(3448)
X(110) = anticomplement of X(125)
X(110) = X(I)-Ceva conjugate of X(J) for these (I,J): (249,6), (250,3)
X(110) = cevapoint of X(I) and X(J) for these (I,J): (3,520), (5,523), (6,512), (141,525)
X(110) = X(I)-cross conjugate of X(J) for these (I,J):
(1,59), (3,250), (6,249), (109,162), (351,111), (512,6), (520,3), (523,54), (526,74)
X(110) = crosssum of X(I) and X(J) for these (I,J): (2,148), (512,647), (520,647)
X(110) = crossdifference of evey pair of points on line X(115)X(125)
X(110) = X(I)-Hirst inverse of X(J) for these (I,J): (1,245), (2,125), (3,246), (4,247)
X(110) = X(I)-beth conjugate of X(J) for these (I,J): (21,759), (643,643)
X(111) = PARRY POINT
Trilinears a/(2a2 - b2 - c2) : b/(2b2 - c2 - a2) : c/(2c2 - a2 - b2)Barycentrics a2/(2a2 - b2 - c2) : b2/(2b2 - c2 - a2) : c2/(2c2 - a2 - b2)
X(111) = Λ(X(2), X(6))
X(111) lies on these lines:
2,99 6,110 23,187 25,112 37,100 42,101 107,393 182,353 230,476 251,827 308,689 352,511 385,892 468,935 512,843 647,842 694,805 931,941
X(111) = reflection of X(1296) in X(3)
X(111) = isogonal conjugate of X(524)
X(111) = isotomic conjugate of X(3266)
X(111) = inverse-in-Brocard-circle of X(353)
X(111) = anticomplement of X(126)
X(111) = cevapoint of X(6) and X(187)
X(111) = X(I)-cross conjugate of X(J) for these (I,J): (23,251), (187,6), (351,110)
X(111) = crossdifference of evey pair of points on line X(351)X(690)
X(112) = Ψ(ORTHOCENTER, SYMMEDIAN POINT)
Trilinears a/(sin 2B - sin 2C) : b/(sin 2C - sin 2A) : c/(sin 2A - sin 2B)= f(a,b,c) : f(b,c,a) : f(c,a,b) where f(a,b,c) = a/[(b2 - c2)(b2 + c2 - a2)]
Barycentrics a2/(sin 2B - sin 2C) : b2/(sin 2C - sin 2A) : c2/(sin 2A - sin 2B)
X(112) = Ψ(X(4), X(6))
X(112) lies on these lines:
2,127 4,32 6,74 19,759 25,111 27,675 28,105 33,609 50,477 54,217 58,103 99,648 100,162 102,284 104,1108 109,163 186,187 230,403 250,691 251,427 286,767 376,577 393,571 523,935 789,811
X(112) = reflection of X(I) in X(J) for these (I,J): (4,132), (1297,3)
X(112) = isogonal conjugate of X(525)
X(112) = anticomplement of X(127)
X(112) = X(I)-Ceva conjugate of X(J) for these (I,J): (249,24), (250,25)
X(112) = cevapoint of X(I) and X(J) for these (I,J): (32,512), (427,523)
X(112) = X(I)-cross conjugate of X(J) for these (I,J): (25,250), (512,4), (523,251)
X(112) = crossdifference of evey pair of points on line X(122)X(125)
X(112) = barycentric product of X(1113) and X(1114)
Centers 113-139
lie on the nine-point circle.
X(113) = JERABEK ANTIPODE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = sin B sin C [(sin C)/(cos C - 2 cos A cos B) + (sin B)/(cos B - 2 cos A cos C)]
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = (sin C)/(cos C - 2 cos A cos B) + (sin B)/(cos B - 2 cos A cos C)
= h(a,b,c) : h(b,c,a) : h(c,a,b),
where h(a,b,c) = b2/(b2SB - 2SASC) + c2/(c2SC - 2SASB),
SA = (b2 + c2 - a2)/2, and SB and SC are defined cyclically (Peter J. C. Moses, 3/2003)
X(113) = nine-point-circle-antipode of X(125)
X(113) = X(74)-of-medial triangle
X(113) = X(104)-of-orthic triangle
X(113) lies on these lines:
2,74 3,122 4,110 5,125 6,13 11,942 52,135 114,690 123,960 127,141 137,546
X(113) = midpoint of X(I) and X(J) for these (I,J): (4,110), (74,146), (265,399), (1553,3258)
X(113) = reflection of X(I) in X(J) for these (I,J): (52,1112), (125,5)
X(113) = complementary conjugate of X(30)
X(113) = X(4)-Ceva conjugate of X(30)
X(113) = crosspoint of X(4) and X(403)
X(113) = crossdifference of any two points on line X(526)X(686)
X(114) = KIEPERT ANTIPODE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[b sec(B + ω) + c sec(C + ω)]= cos(B - C) cos 2ω - sin ω sin(A + ω) (Peter J. C. Moses, 9/12/03)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b sec(B + ω) + c sec(C + ω)
X(114) = nine-point-circle-antipode of X(115)
X(114) = X(98)-of-medial triangle
X(114) = X(103)-of-orthic triangle.
X(114) lies on these lines:
2,98 3,127 4,99 5,39 25,135 52,211 113,690 132,684 136,427 325,511 381,543
X(114) = isogonal conjugate of X(2065)
X(114) = midpoint of X(I) and X(J) for these (I,J): (4,99), (98,147)
X(114) = reflection of X(I) in X(J) for these (I,J): (3,620), (115,5)
X(114) = complementary conjugate of X(511)
X(114) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,230), (4,511)
X(114) = crosspoint of X(2) and X(325)
X(114) = orthojoin of X(230)
X(115) = CENTER OF KIEPERT HYPERBOLA
Trilinears bc(b2 - c2)2 : ca(c2 - a2)2 : ab(a2 - b2)2= cos A - 2 cos(B - C) + cot ω sin A (Peter J. C. Moses, 9/12/03)
Barycentrics (b2 - c2)2 : (c2 - a2)2 : (a2 - b2)2
If you have The Geometer's Sketchpad, you can view Kiepert Hyperbola, showing X(115).
X(115) lies on the nine-point circle
X(115) = X(99)-of-medial triangle
X(115) = X(101)-of-orthic triangle
Roland H. Eddy and R. Fritsch, "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," Mathematics Magazine 67 (1994) 188-205.
X(115) lies on these lines:
2,99 4,32 5,39 6,13 11,1015 30,187 50,231 53,133 76,626 116,1086 120,442 125,245 127,338 128,233 129,389 131,216 232,403 316,385 325,538 395,530 396,531 593,1029 804,1084
X(115) = midpoint of X(I) and X(J) for these (I,J): (4,98), (13,14), (99,148), (316,385)
X(115) = reflection of X(I) in X(J) for these (I,J): (99,620), (114,5), (187,230), (325,625)
X(115) = isogonal conjugate of X(249)
X(115) = inverse-in-orthocentroidal-circle of X(6)
X(115) = complementary conjugate of X(512)
X(115) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,523), (4,512), (338,125)
X(115) = crosspoint of X(I) and X(J) for these (I,J): (2,523), (68,525)
X(115) = crosssum of X(I) and X(J) for these (I,J): (6,110), (24,112), (163,849)
X(115) = crossdifference of evey pair of points on line X(110)X(351)
X(115) = X(2)-Hirst inverse of X(148)
X(116) = MIDPOINT OF X(4) AND X(103)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b) wheref(a,b,c) = bc[(b - c)2(b2 + bc + c2 - ab - ac)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where
g(a,b,c) = (b - c)2(b2 + bc + c2 - ab - ac)
X(116) lies on the nine-point circle
X(116) = X(101)-of-medial triangle.
X(116) lies on these lines: 2,101 4,103 5,118 10,120 115,1086 119,142 121,141 124,928
X(116) = midpoint of X(I) and X(J) for these (I,J): (4,103), (101,150)
X(116) = reflection of X(118) in X(5)
X(116) = complementary conjugate of X(514)
X(116) = X(4)-Ceva conjugate of X(514)
X(117) = MIDPOINT OF X(4) AND X(109)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = g(b,c,a) + g(c,b,a), andg(b,c,a) = b2c/[c(sec B - sec C) + a(sec B - sec A)]
Barycentrics h(a,b,c) : h(b,c,a) : h(c,a,b) where h(a,b,c) = af(a,b,c)
X(117) lies on the nine-point circle
X(117) = X(102)-of-medial triangle.
X(117) lies on these lines: 2,102 4,109 5,124 10,123 11,65 118,928 136,407
X(117) = midpoint of X(I) and X(J) for these (I,J): (4,109), (102,151)
X(117) = reflection of X(124) in X(5)
X(117) = complementary conjugate of X(515)
X(117) = X(4)-Ceva conjugate of X(515)
X(118) = MIDPOINT OF X(4) AND X(101)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = g(b,c,a) + g(c,b,a), andg(b,c,a) = b3c/[(b - c) cot A + (b - a) cot C]
Barycentrics h(a,b,c) : h(b,c,a) : h(c,a,b) where h(a,b,c) = af(a,b,c)
X(118) lies on the nine-point circle
X(118) = X(103)-of-medial triangle.
X(118) lies on these lines: 2,103 4,101 5,116 11,226 117,928 122,440 136,430 381,544 516,910
X(118) = midpoint of X(I) and X(J) for these (I,J): (4,101), (103,152)
X(118) = reflection of X(116) in X(5)
X(118) = complementary conjugate of X(516)
X(118) = X(4)-Ceva conjugate of X(516)
X(119) = FEUERBACH ANTIPODE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (csc A)(-1 + cos B + cos C)[sin 2B + sin 2C + 2(-1 + cos A)(sin B + sin C)]
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B) where
g(A,B,C) = (-1 + cos B + cos C)[sin 2B + sin 2C + 2(-1 + cos A)(sin B + sin C)]
X(119) = nine-point-circle-antipode of X(11)
X(119) = X(104)-of-medial triangle.
X(119) lies on these lines:
1,5 2,104 3,123 4,100 10,124 116,142 125,442 135,431 136,429 214,515 381,528 517,908
X(119) = midpoint of X(I) and X(J) for these (I,J): (4,100), (104,153)
X(119) = reflection of X(11) in X(5)
X(119) = complement of X(104)
X(119) = complementary conjugate of X(517)
X(119) = X(4)-Ceva conjugate of X(517)
X(120) = X(105)-OF-MEDIAL-TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc[2abc - (b + c)(a2 + (b - c)2)](b2 + c2 - ab -ac)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = [2abc - (b + c)(a2 + (b - c)2)](b2 + c2 - ab -ac)
X(120) lies on the nine-point circle
X(120) = X(105)-of-medial triangle.
X(120) lies on these lines: 2,11 10,116 12,85 115,442
X(120) = complementary conjugate of X(518)
X(120) = X(4)-Ceva conjugate of X(518)
X(121) = X(106)-OF-MEDIAL-TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc(b + c - 2a)[b3 + c3 + a(b2 + c2) - 2bc(b + c)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = (b + c - 2a)[b3 + c3 + a(b2 + c2) - 2bc(b + c)]
X(121) lies on the nine-point circle
X(121) = X(106)-of-medial triangle.
X(121) lies on these lines: 2,106 10,11 116,141
X(121) = complementary conjugate of X(519)
X(121) = X(4)-Ceva conjugate of X(519)
X(122) = X(107)-OF-MEDIAL-TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (b2 - c2)2(cos A - cos B cos C) cot2A
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = a(b2 - c2)2(cos A - cos B cos C) cot2A
X(122) lies on the nine-point circle
X(122) = X(107)-of-medial triangle.
X(122) lies on these lines: 2,107 3,113 5,133 118,440 125,684 138,233
X(122) = reflection of X(133) in X(5)
X(122) = complementary conjugate of X(520)
X(122) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,520), (253,525)
X(122) = crosssum of X(I) and X(J) for these (I,J): (64,1301), (112,154)
X(122) = crosspoint of X(253) and X(525)
X(122) = crossdifference of evey pair of points on line X(112)X(1301)
X(123) = X(108)-OF-MEDIAL-TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (csc A)(sec B - sec C)[(sec A)(sin2B - sin2C) + sin C tan C - sin B tan B]
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B) where
g(A,B,C) = (sec B - sec C)[(sec A)(sin2B - sin2C) + sin C tan C - sin B tan B]
X(123) lies on the nine-point circle
X(123) = X(108)-of-medial triangle.
X(123) lies on these lines: 2,108 3,119 10,117 113,960
X(123) = complementary conjugate of X(521)
X(123) = X(4)-Ceva conjugate of X(521)
X(124) = X(109)-OF-MEDIAL-TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc(b + c - a)(b - c)2[(b + c)(b2 + c2 - a2 - bc) + abc]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = (b + c - a)(b - c)2
[(b + c)(b2 + c2 - a2 - bc) + abc]
X(124) lies on the nine-point circle
X(124) = X(109)-of-medial triangle.
X(124) lies on these lines: 2,109 4,102 5,117 10,119 116,928
X(124) = midpoint of X(4) and X(102)
X(124) = reflection of X(117) in X(5)
X(124) = complementary conjugate of X(522)
X(124) = X(4)-Ceva conjugate of X(522)
X(125) = CENTER OF JERABEK HYPERBOLA
Trilinears cos A sin2(B - C) : cos B sin2(C - A) : cos C sin2(A - B)= (sec A)(c cos C - b cos B)2 : (sec B)(a cos A - c cos C)2 : (sec C)(b cos B - a cos A)2
= bc(b2 + c2 - a2)(b2 - c2)2 : ca(c2 + a2 - b2)(c2 - a2)2 : ab(a2 + b2 - c2)(a2 - b2)2
Barycentrics (sin 2A)[sin(B - C)]2 : (sin 2B)[sin(C - A)]2 : (sin 2C)[sin(A - B)]2
X(125) lies on the nine-point circle
X(125) = X(110)-of-medial triangle
X(125) = X(100)-of-orthic triangle, if ABC is acute
Roland H. Eddy and R. Fritsch, "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," Mathematics Magazine 67 (1994) 188-205.
X(125) lies on these lines:
2,98 3,131 4,74 5,113 6,67 51,132 68,1092 69,895 115,245 119,442 122,684 126,141 128,140 136,338 381,541 511,858
X(125) = midpoint of X(I) and X(J) for these (I,J): (3,265), (4,74), (6,67), (110,3448)
X(125) = reflection of X(I) in X(J) for these (I,J): (113,5), (185,974), (1495,468), (1511,140), (1539,546)
X(125) = isogonal conjugate of X(250)
X(125) = inverse-in-Brocard-circle of X(184)
X(125) = complement of X(110)
X(125) = complementary conjugate of X(523)
X(125) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,523), (66,512), (68,520), (69,525), (338,115)
X(125) = crosspoint of X(I) and X(J) for these (I,J): (4,523), (69,525), (338,339)
X(125) = crosssum of X(I) and X(J) for these (I,J): (3,110), (25,112), (162,270), (1113,1114)
X(125) = crossdifference of evey pair of points on line X(110)X(112)
X(125) = X(115)-Hirst inverse of X(868)
X(125) = X(2)-line conjugate of X(110)
X(126) = X(111)-OF-MEDIAL-TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc(2a2 - b2 - c2)[b4 + c4 + a2(b2 + c2) - 4b2c2]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = (2a2 - b2 - c2)[b4 + c4 + a2(b2 + c2) - 4b2c2]
X(126) lies on the nine-point circle
X(126) = X(111)-of-medial triangle.
X(126) lies on these lines: 2,99 125,141 625,858
X(126) = complement of X(111)
X(126) = complementary conjugate of X(524)
X(126) = X(4)-Ceva conjugate of X(524)
X(127) = X(112)-OF-MEDIAL-TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc(sin 2B - sin 2C)[(b2 - c2)sin 2A - b2sin 2B + c2sin 2C]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = (sin 2B - sin 2C)[(b2 - c2)sin 2A - b2sin 2B + c2sin 2C]
X(127) lies on the nine-point circle
X(127) = X(112)-of-medial triangle.
X(127) lies on these lines: 2,112 3,114 5,132 113,141 115,338 133,381 125,140
X(127) = reflection of X(132) in X(5)
X(127) = complement of X(112)
X(127) = anticomplementary conjugate of X(525)
X(127) = X(4)-Ceva conjugate of X(525)
X(128) = X(74)-OF-ORTHIC-TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)(cos 2B + cos 2C)(1 + 2 cos 2A)(cos 2A + 2 cos 2B cos 2C)
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B) where g(A,B,C) = (sin A) f(A,B,C)
X(128) lies on the nine-point circle
X(128) = X(74)-of-orthic triangle.
X(128) lies on these lines: 5,137 52,134 53,139 115,233 125,140
X(128) = reflection of X(137) in X(5)
X(128) = X(2)-Ceva conjugate of X(231)
X(128) = orthojoin of X(231)
X(129) = X(98)-OF-ORTHIC-TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)(sin 2A)(sin 2B + sin 2C) s(A,B,C) t(A,B,C),
s(A,B,C) = sin4(2B) + sin4(2C) - sin2(2A) sin2(2B) - sin2(2A) sin2(2C),
t(A,B,C) = sin4(2A) + sin2(2A) u(A,B,C) + v(A,B,C),
u(A,B,C) = sin 2B sin 2C - sin2(2B) - sin2(2C),
v(A,B,C) = (sin 2B sin 2C)(sin 2B - sin 2C)2
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B) where g(A,B,C) = (sin A) f(A,B,C)
X(129) lies on the nine-point circle
X(129) = X(98)-of-orthic triangle.
X(129) lies on these lines: 5,130 51,137 52,139 115,389
X(129) = reflection of X(130) in X(5)
X(130) = X(99)-OF-ORTHIC-TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sin A)(sin 2B + sin 2C)[(sin 2B - sin 2C)2][sin2(2A) + sin 2B sin 2C]
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(130 lies on the nine-point circle
X(130) = X(99)-of-orthic triangle
X(130) lies on these lines: 5,129 51,138
X(130) = reflection of X(129) in X(5)
X(131) = INTERSECTION OF LINES X(3)X(125) AND X(4)X(135)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)[2T - S(sec 2B + sec 2C)](T - S sec 2A),
S = sin 2A + sin 2B + sin 2C, T = tan 2A + tan 2B + tan 2C
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(131) lies on the nine-point circle
X(131) = X(102)-of-orthic triangle if ABC is acute.
X(131) lies on these lines: 3,125 4,135 5,136 115,216
X(131) = reflection of X(136) in X(5)
X(132) = INTERSECTION OF LINES X(2)X(107) AND X(4)X(32)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A) u(A,B,C) v(A,B,C),
u(A,B,C) = [sin2(2A) + (sin 2B - sin 2C)2 + (sin 2A)(sin 2A - sin 2B - sin 2C)],
v(A,B,C) = [sin2(2B) + sin2(2C) - (sin 2A sin 2B) - (sin 2A sin 2C)]
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(132) lies on the nine-point circle
X(132) = X(105)-of-orthic triangle if ABC is acute
X(132) lies on these lines: 2,107 4,32 5,127 25,136 51,125 114,684 137,428 147,648
X(132) = midpoint of X(4) and X(112)
X(132) = reflection of X(127) in X(5)
X(132) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,232), (4,1503)
X(132) = X(4)-line conjugate of X(248)
X(132) = crossdifference of evey pair of points on line X(248)X(684)
X(133) = INTERSECTION OF LINES X(4)X(74) AND X(5)X(122)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A) u(A,B,C) v(A,B,C),
u(A,B,C) = (sin 2B - sin 2C)2 + sin 2A sin 2B - sin 2A sin 2C - 2 sin 2B sin 2C,
v(A,B,C) = 2 sin 2A - sin 2B - sin 2C
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(133) lies on the nine-point circle
X(133) = X(106)-of-orthic triangle is ABC is acute.
X(133) lies on these lines: 4,74 5,122 53,115 127,381 136,235
X(133) = midpoint of X(4) and X(107)
X(133) = reflection of X(122) in X(5)
X(134) = X(107)-OF-ORTHIC-TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A) u(A,B,C) [v(B,C,A) - v(C,B,A)],
u(A,B,C) = (sin 2A)[sin2(2B) - sin2(2C)][sin2(2B) + sin2(2C) - sin2(2A)]2,
v(B,C,A) = sin 2C [sin2(2A) - sin2(2B)][sin2(2A) + sin2(2B) - sin2(2C)]2
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(134) lies on the nine-point circle
X(134) = X(107)-of-orthic triangle
X(134) lies on this line: 52,128
X(135) = INTERSECTION OF LINE X(4)X(131) AND X(25)X(114)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)[(sin 2B)/(sec 2C - sec 2A) + (sin 2C)/(sec 2A - sec 2B)]
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = (tan A)[(sin 2B)/(sec 2C - sec 2A) + (sin 2C)/(sec 2A - sec 2B)]
X(135) lies on the nine-point circle
X(135) = X(108)-of-orthic triangle if ABC is acute
X(135) lies on these lines: 4,131 25,114 52,113 119,431
X(136) = INTERSECTION OF LINE X(4)X(110) AND X(25)X(132)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)[(sin 2B - sin 2C)2](sin 2B + sin 2C - sin 2A) u(A,B,C),
u(A,B,C) = [sin2(2B) + sin2(2C) - sin2(2A)]
Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
X(136) lies on the nine-point circle
X(136) =X(109)-of-orthic triangle if ABC is acute
X(136) lies on these lines:
2,925 4,110 5,131 25,132 68,254 114,427 117,407 118,430 119,429 125,338 127,868 133,235
X(136) = reflection of X(131) in X(5)
X(136) = complement of X(925)
X(136) = complementary conjugate of X(924)
X(136) = X(254)-Ceva conjugate of X(523)
X(137) = X(110)-OF-ORTHIC-TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)(sin 2B + sin 2C)[(sin 2B - sin 2C)2] u(A,B,C),
u(A,B,C) = [sin2(2A) - sin2(2B) - sin2(2C) - sin 2B sin 2C]
Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
X(137) lies on the nine-point circle
X(137) = X(110)-of-orthic triangle
X(137) lies on these lines: 5,128 51,129 53,138 113,546 132,428
X(137) = reflection of X(128) in X(5)
X(137) = complement of X(930)
X(137) = X(4)-Ceva conjugate of X(1510)
X(137) = crosssum of X(252) and X(930)
X(138) = X(111)-OF-ORTHIC-TRIANGLE
Trilinears (v + w) sec A : (w + u) sec B : (u + v) sec C, whereu = u(A,B,C) = (sin 2A)/(2 sin22A - sin22B - sin22C), v = u(B,C,A), w = u(C,A,B)
Barycentrics (v + w) tan A : (w + u) tan B : (u + v) tan C
X(138) lies on the nine-point circle
X(138) = X(111)-of-orthic triangle
X(138) lies on these lines: 51,130 53,137 122,233
X(139) = X(112)-OF-ORTHIC-TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)(sin 2B + sin 2C)[(sin 2B - sin 2C)2][sin2(2B) + sin2(2C) - sin2(2A)] u(A,B,C),
u(A,B,C) = (sin 2B)4 + (sin 2C)4 - (sin 2A)4 + (sin 2B sin 2C)[sin2(2B) + sin2(2C) - sin2(2A)]
Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
X(139) lies on the nine-point circle
X(139) = X(112) of the orthic triangle
X(139) lies on these lines: 52,129 53,128
Centers 140- 170
113- 127, 140- 143: centers of the medial triangle
128- 139: centers of the orthic triangle
144- 153: centers of the anticomplementary triangle
154- 157, 159- 163: centers of the tangential triangle
164- 170: centers of the excentral triangle
X(140) = MIDPOINT OF X(3) AND X(5)
Trilinears 2 cos A + cos(B - C) : 2 cos B + cos(C - A) : 2 cos C + cos(A - B)= cos A + 2 sin B sin C : cos B + 2 sin C sin A : cos C + 2 sin A sin B
= 3 cos A + 2 cos B cos C : 3 cos B + 2 cos C cos A : 3 cos C + 2 cos A cos B
= 2 sec A + 3 sec B sec C : 2 sec B + 3 sec C sec A : 2 sec C + 3 sec A sec B
= f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = bc[b cos(C - A) + c cos(B - A)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = b cos(C - A) + c cos(B - A)
X(140) midpoint of X(I) and X(J) for these (I,J): (3,5), (2883,3357)
X(140) = crosspoint of the two Napoleon points
X(140) = X(5)-of-medial triangle
X(140) lies on these lines:
2,3 10,214 11,35 12,36 15,18 16,17 39,230 54,252 55,496 56,495 61,395 62,396 95,340 125,128 141,182 143,511 195,323 298,628 299,627 302,633 303,634 343,569 371,615 372,590 524,575 576,597 601,748 602,750 618,630 619,629
X(140) is the {X(2),X(3)}-harmonic conjugate of X(5). For a list of other harmonic conjugates of X(140), click More at the top of this page.
X(140) = midpoint of X(I) and X(J) for these (I,J): (3,5), (141,182)
X(140) = reflection of X(I) in X(J) for these (I,J): (546,5), (547,2), (548,3)
X(140) = inverse-in-orthocentroidal-circle of X(1656)
X(140) = isogonal conjugate of X(1173)
X(140) = complement of X(5)
X(140) = complementary conjugate of X(1209)
X(140) = X(2)-Ceva conjugate of X(233)
X(140) = crosspoint of X(I) and X(J) for these (I,J): (2,95), (17,18)
X(140) = crosssum of X(I) and X(J) for these (I,J): (6,51), (61,62)
X(141) = COMPLEMENT OF SYMMEDIAN POINT
Trilinears bc(b2 + c2) : ca(c2 + a2) : ab(a2 + b2)= csc2A sin(A + ω) : csc2B sin(B + ω) : csc2C sin(C + ω)
Barycentrics b2 + c2 : c2 + a2 : a2 + b2
X(141) = X(6)-of-medial triangle
X(141) lies on these lines:
2,6 3,66 5,211 10,142 37,742 39,732 45,344 53,264 67,110 75,334 76,698 95,287 99,755 113,127 116,121 125,126 140,182 239,319 241,307 308,670 311,338 317,458 320,894 384,1031 441,577 498,611 499,613 523,882 542,549 575,629 997,1060
X(141) is the {X(2),X(69)}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(141), click More at the top of this page.
X(141) = midpoint of X(I) and X(J) for these (I,J): (1,3416), (6,69), (8,3242), (66,159), (67,110), (69,3313), (1843,3313) (2930, 3448)
X(141) = reflection of X(I) in X(J) for these (I,J): (182,140), (597,2), (1353,575), (1386,1125)
X(141) = isogonal conjugate of X(251)
X(141) = isotomic conjugate of X(83)
X(141) = inverse-in-nine-point-circle of X(625)
X(141) = complement of X(6)
X(141) = complementary conjugate of X(2)
X(141) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,39), (67,524), (110,525)
X(141) = X(39)-cross conjugate of X(427)
X(141) = crosspoint of X(2) and X(76)
X(141) = crosssum of X(6) and X(32)
X(141) = X(39)-Hirst inverse of X(732)
X(141) = X(645)-beth conjugate of X(141)
X(142) = COMPLEMENT OF MITTENPUNKT
Trilinears b + c - [(b - c)2]/a : c + a - [(c - a)2]/b : a + b - [(a - b)2]/cBarycentrics ab + ac - (b - c)2 : bc + ba - (c - a)2 : ca + cb - (a - b)2
X(142) = X(9)-of- medial triangle
X(142) = centroid of the set {X(1), X(4), X(7), X(40)}
X(142) lies on these lines: 1,277 2,7 3,516 5,971 10,141 37,1086 86,284 116,119 214,528 269,948 377,950 474,954
X(142) is the {X(2),X(7)}-harmonic conjugate of X(9). For a list of other harmonic conjugates, click More at the top of this page.
X(142) = midpoint of X(I) and X(J) for these (I,J): (7,9), (8,3243), (100,3254)
X(142) = reflection of X(1001) in X(1125)
X(142) = isogonal conjugate of X(1174)
X(142) = complement of X(9)
X(142) = complementary conjugate of X(3452)
X(142) = X(100)-Ceva conjugate of X(514)
X(142) = crosspoint of X(2) and X(85)
X(142) = crosssum of X(6) and X(41)
X(142) = X(190)-beth conjugate of X(142)
X(143) = NINE-POINT CENTER OF ORTHIC TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)[cos(2C - 2A) + cos(2A - 2B)]
Trilinears g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = [1 - 2 cos(2A)]cos(B - C)]
Trilinears h(A,B,C) : h(B,C,A) : h(C,A,B), where
h(A,B,C) = sec A cos(3A) cos(B - C) (Manol Iliev, 4/01/07)
Barycentrics k(A,B,C) : k(B,C,A) : k(C,A,B), where k(A,B,C) = (tan A)[cos(2C - 2A) + cos(2A - 2B)]
This point is the third of three Spanish Points developed by Antreas P. Hatzipolakis and Javier Garcia Capitan in 2009; see X(3567).
X(143) = X(5)-of-orthic triangle, if ABC is acute.
X(143) = insimilicenter of the circumcircle and the nine-point circle of the orthic triangle. (Peter Moses, July 1, 2009)
X(143) lies on these lines: 4,94 5,51 6,26 25,156 30,389 110,195 140,511 324,565
X(143) is the {X(51),X(52)}-harmonic conjugate of X(5). For a list of other harmonic conjugates, click More at the top of this page.
X(143) = midpoint of X(5) and X(52)
X(143) = isogonal conjugate of X(252)
X(143) = X(137)-cross conjugate of X(1510)
X(144) = ANTICOMPLEMENT OF X(7)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = (csc A)(tan B/2 + tan C/2 - tan A/2)
Barycentrics tan B/2 + tan C/2 - tan A/2 : tan C/2 + tan A/2 - tan B/2 : tan A/2 + tan B/2 - tan C/2
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b),
where g(a,b,c) = 1/(a - b - c) + 1/(a - b + c) + 1/(a + b - c) (Peter J. C. Moses, 4/8/03)
X(144) = X(7)-of-anticomplementary triangle
X(144) lies on these lines:
2,7 8,516 20,72 21,954 69,190 75,391 100,480 145,192 219,347 220,279 320,344
X(144) is the {X(7),X(9)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(144), click More at the top of this page.
X(144) = reflection of X(I) in X(J) for these (I,J): (7,9), (145,390), (149,1156)
X(144) = anticomplement of X(7)
X(144) = anticomplementary conjugate of X(3434)
X(144) = X(8)-Ceva conjugate of X(2)
X(144) = X(I)-beth conjugate of X(J) for these (I,J): (190,144), (645,346)
X(145) = ANTICOMPLEMENT OF NAGEL POINT
Trilinears bc(3a - b - c) : ca(3b - c - a) : ab(3c - a - b)Barycentrics 3a - b - c : 3b - c -a : 3c - a - b
X(145) = X(8)-of-anticomplementary triangle
X(145) lies on these lines: 1,2 4,149 6,346 20,517 21,956 37,391 56,100 72,452 81,1043 144,192 218,644 279,664 329,950 330,1002 377,1056 404,999 515,962
X(145) is the {X(1),X(8)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(145), click More at the top of this page.
X(145) = midpoint of X(2) and X(3241)
X(145) = reflection of X(I) in X(J) for these (I,J): (3,1483), (4,1482), (8,1), (20,944), (100,1317), (144,390), (149,1320)
X(145) = isogonal conjugate of X(3445)
X(145) = anticomplement of X(8)
X(145) = anticomplementary conjugate of X(3436)
X(145) = X(7)-Ceva conjugate of X(2)
X(145) = crosssum of X(663) and X(1015)
X(145) = X(643)-beth conjugate of X(56)
X(146) = REFLECTION OF X(20) IN X(110)
Trilinears bc(-avw + bwu + cuv) : ca(-bwu + cuv + avw) : ab(-cuv + avw + bwu), whereu = u(A,B,C) = cos A - 2 cos B cos C, v = u(B,C,A), w = u(C,A,B)
Barycentrics -avw + bwu + cuv : -bwu + cuv + avw : -cuv + avw + bwu
X(146) = X(74)-of-anticomplementary triangle
X(146) lies on these lines: 2,74 4,94 20,110 30,323 147,690 148,193
X(146) is the {X(74),X(113)}-harmonic conjugate of X(2). For a list of harmonic conjugates, click More at the top of this page.
X(146) = reflection of X(I) in X(J) for these (I,J): (20,110), (74,113), (265,1539)
X(146) = anticomplementary conjugate of X(30)
X(147) = TARRY POINT OF ANTICOMPLEMENTARY TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc[a8 + (b2 + c2)a6 - (2b4 + 3b2c2 + 2c4)a4
+ (b6 + b4c2 + b2c4 + c6)a2 - b8 + b6c2 + b2c6 - c8]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(147) = X(98)-of-anticomplementary triangle
X(147) lies on these lines: 1,150 2,98 4,148 20,99 132,648 146,690 684,804
X(147) is the {X(98),X(114)}-harmonic conjugate of X(2). For a list of harmonic conjugates, click More at the top of this page.
X(147) = reflection of X(I) in X(J) for these (I,J): (20,99), (98,114), (148,4), (385,1513)
X(147) = anticomplementary conjugate of X(511)
X(147) = X(325)-Ceva conjugate of X(2)
X(148) = STEINER POINT OF ANTICOMPLEMENTARY TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc[a4 - (b2 - c2)2 + b2c2 - a2b2 - a2c2]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a4 - (b2 - c2)2 + b2c2 - a2b2 - a2c2
X(148) = X(99)-of-anticomplementary triangle
X(148) lies on these lines: 2,99 4,147 13,617 20,98 30,385 146,193 316,538
X(148) = reflection of X(I) in X(J) for these (I,J): (2,671), (20,98), (99,115), (147,4), (616,14), (617,13)
X(148) = anticomplementary conjugate of X(512)
X(148) = X(523)-Ceva conjugate of X(2)
X(148) = X(2)-Hirst inverse of X(115)
X(149) = REFLECTION OF X(20) IN X(104)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc[b3 + c3 - a3 + (a2 - bc)(b + c) + a(bc - b2 - c2)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = b3 + c3 - a3 + (a2 - bc)(b + c) + a(bc - b2 - c2)
X(149) = X(100)-of-anticomplementary triangle
X(149) lies on these lines: 2,11 4,145 8,80 20,104 151,962 377,1058 404,496
X(149) is the {X(11),X(100)}-harmonic conjugate of X(2). For a list of harmonic conjugates, click More at the top of this page.
X(149) = reflection of X(I) in X(J) for these (I,J): (3,1484), (8,80), (20,104), (100,11), (144,1156), (145,1320), (153,4)
X(149) = isogonal conjugate of X(3446)
X(149) = anticomplementary conjugate of X(513)
X(150) = REFLECTION OF X(20) IN X(103)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc[b4 + c4 - a4 + a(bc2 +cb2 - b3 - c3) - bc(a2 + b2 + c2) + (b + c)a3]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = b4 + c4 - a4 + a(bc2 +cb2 - b3 - c3) - bc(a2 + b2 + c2) + (b + c)a3
X(150) = X(101)-of-anticomplementary triangle
X(150) lies on these lines: 1,147 2,101 4,152 7,80 20,103 69,668 85,355 295,334 348,944 664,952
X(150) = reflection of X(I) in X(J) for these (I,J): (20,103), (101,116), (152,4), (664,1565)
X(150) = anticomplementary conjugate of X(514)
X(151) = REFLECTION OF X(20) IN X(109)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (by + cz - ax)/a, where x : y : z = X(102)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(151) = X(102)-of-anticomplementary triangle
X(151) lies on these lines: 2,102 20,109 149,962 152,928
X(151) = reflection of X(I) in X(J) for these (I,J): (20,109), (102,117)
X(151) = anticomplementary conjugate of X(515)
X(152) = REFLECTION OF X(20) IN X(101)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (by + cz - ax)/a, where x : y : z = X(103)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(152) = X(103)-of-anticomplementary triangle
X(152) lies on these lines: 2,103 4,150 20,101 151,928
X(152) = reflection of X(I) in X(J) for these (I,J): (20,101), (103,118), (150,4)
X(152) = anticomplementary conjugate of X(516)
X(153) = REFLECTION OF X(20) IN X(100)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (by + cz - ax)/a, where x : y : z = X(104)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(153) = X(104)-of-anticomplementary triangle
X(153) lies on these lines: 2,104 4,145 7,80 11,388 20,100 515,908
X(153) = reflection of X(I) in X(J) for these (I,J): (20,100), (104,119), (149,4), (1320,1537)
X(153) = anticomplementary conjugate of X(517)
X(154) = X(3)-CEVA CONJUGATE OF X(6)
Trilinears (cos A - cos B cos C)a2 : (cos B - cos C cos A)b2 : (cos C - cos A cos B)c2= a(tan B + tan C - tan A) : b(tan C + tan A - tan B): c(tan A + tan B - tan C)
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin2 A)(tan B + tan C - tan A)
X(154) = X(2)-of-tangential triangle
X(154) lies on these lines:
3,64 6,25 22,110 26,155 31,56 48,55 160,418 197,692 198,212 205,220 237,682
X(154) is the {X(26),X(156)}-harmonic conjugate of X(155). For a list of harmonic conjugates of X(154), click More at the top of this page.
X(154) = isogonal conjugate of X(253)
X(154) = X(3)-Ceva conjugate of X(6)
X(154) = crosssum of X(I) and X(J) for these (I,J): (64,1073), (122,525)
X(154) = X(109)-beth conjugate of X(154)
X(155) EIGENCENTER OF ORTHIC TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)[cos2B + cos2C - cos2A]Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)
X(4) of the tangential triangle. This point is also the center of the circle which cuts (extended) lines BC, CA, AB in pairs of points A' and A", B' and B", C' and C", respectively, such that angles A'AA", B'BB", C'CC" are all right angles. This is the Dou circle, described in
Jordi Dou, Problem 1140, Crux Mathematicorum, 28 (2002) 461-462.
X(155) lies on these lines:
1,90 3,49 4,254 5,6 20,323 24,110 25,52 26,154 30,1498 159,511 195,381 382,399 450,1075 648,1093 651,1068
X(155) is the {X(26),X(156)}-harmonic conjugate of X(154). For a list of harmonic conjugates of X(155), click More at the top of this page.
X(155) = reflection of X(I) in X(J) for these (I,J): (3,1147), (26,156), (68,5)
X(155) = isogonal conjugate of X(254)
X(155) = eigencenter of cevian triangle of X(4)
X(155) = eigencenter of anticevian triangle of X(3)
X(155) = X(4)-Ceva conjugate of X(3)
X(155) = crosssum of X(136) and X(523)
X(156) = X(5)-OF-TANGENTIAL-TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b2y/v + c2z/w - a2x/u],u = u(A,B,C) = sin 2A, v = u(B,C,A), w = u(C,A,B);
x = x(A,B,C) = u2(v2 + w2) - (v2 - w2)2, y = x(B,C,A), z = x(C,A,B)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(156) = X(5)-of-tangential triangle
X(156) lies on these lines: 3,74 4,49 5,184 25,143 26,154 54,381 546,578 550,1092
X(156) is the {X(154),X(155)}-harmonic conjugate of X(26). For a list of harmonic conjugates, click More at the top of this page.
X(156) = midpoint of X(26) and X(155)
X(157) = X(6)-OF-TANGENTIAL-TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b3cos B + c3cos C - a3cos A]= g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = a[a6 - b6 - c6 - a4(b2 + c2) + b4(c2 + a2) + c4(a2 + b2)]
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(157) = X(6)-of-tangential triangle
X(157) lies on these lines: 3,66 6,248 22,183 25,53 161,418 206,216
X(157) = X(264)-Ceva conjugate of X(6)
X(157) = crosssum of X(127) and X(520)
X(158) = X(19)-CROSS CONJUGATE OF X(92)
Trilinears sec2A : sec2B : sec2C= 1/(1 + cos 2A) : 1/(1 + cos 2B) : 1/(1 + cos 2C)
Barycentrics sec A tan A : sec B tan B : sec C tan C
X(158) lies on these lines:
1,29 3,243 4,65 10,318 37,281 46,412 47,162 75,240 107,759 225,1093 255,775 286,969 823,897 920,921
X(158) = isogonal conjugate of X(255)
X(158) = isotomic conjugate of X(326)
X(158) = X(I)-cross conjugate of X(J) for these (I,J): (19,92), (225,4)
X(158) = crosssum of X(520) and X(1364)
X(158) = crossdifference of evey pair of points on line X(680)X(822)
X(158) = X(I)-aleph conjugate of X(J) for these (I,J): (821,158), (1105,255)
X(158) = X(107)-beth conjugate of X(34)
X(159) = X(9)-OF-TANGENTIAL-TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = a[(a2 + b2 + c2)sin 2A + (c2 - b2 - a2)sin 2B + (b2 - c2 - a2)sin 2C]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(159) = X(9)-of-tangential triangle
X(159) lies on these lines: 3,66 6,25 22,69 23,193 155,511 197,200
X(159) = reflection of X(I) in X(J) for these (I,J): (6,206), (66,141)
X(159) = X(I)-Ceva conjugate of X(J) for these (I,J): (22,3), (69,6)
X(159) = crosssum of X(127) and X(523)
X(160) = X(37)-OF-TANGENTIAL-TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = a[(b2 + c2)sin 2A + (c2 - a2)sin 2B + (b2 - a2)sin 2C]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(160) = X(37)-of-tangential triangle
X(160) lies on these lines: 3,66 6,237 22,325 95,327 154,418 206,57
X(160) = X(95)-Ceva conjugate of X(6)
X(160) = crosssum of X(338) and X(512)
X(161) = X(63)-OF-TANGENTIAL-TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = a[(a2 + b2 + c2)sin2(2A) + (c2 - b2 - a2)sin2(2B) + (b2 - c2 - a2)sin2(2C)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(161) = X(63)-of-tangential triangle
X(161) lies on these lines: 6,25 22,343 26,68 157,418
X(161) = X(68)-Ceva conjugate of X(6)
X(162) = CEVAPOINT OF X(108) AND X(109)
Trilinears 1/(sin 2B - sin 2C) : 1/(sin 2C - sin 2A) : 1/(sin 2A - sin 2B)= f(a,b,c) : f(b,c,a) : f(c,a,b) , where f(a,b,c) = 1/[(b2 - c2)(b2 + c2 - a2)]
Barycentrics a/(sin 2B - sin 2C) : b/(sin 2C - sin 2A) : c/(sin 2A - sin 2B)
X(162) lies on these lines:
4,270 6,1013 19,897 27,673 28,88 29,58 31,92 47,158 63,204 100,112 107,109 108,110 190,643 238,415 240,896 242,422 255,1099 412,580 799,811
X(162) = isogonal conjugate of X(656)
X(162) = X(250)-Ceva conjugate of X(270)
X(162) = cevapoint of X(I) and X(J) for this (I,J): (108,109)
X(162) = X(I)-cross conjugate of X(J) for these (I,J): (108,107), (109,110)
X(162) = crosssum of X(810) and X(822)
X(162) = X(I)-aleph conjugate of X(J) for these (I,J): (28,1052), (107,920), (162,1), (648,63)
X(162) = trilinear pole of line X(1)X(19)
X(162) = trilinear product of X(1113) and X(1114)
X(163) = TRILINEAR PRODUCT X(6)*X(110)
Trilinears (sin 2A)/(sin 2B - sin 2C) : (sin 2B)/(sin 2C - sin 2A) : (sin 2C)/(sin 2A - sin 2B)= a2/(b2 - c2) : b2/(c2 - a2) : c2/(a2 - b2)
Barycentrics a3/(b2 - c2) : b3/(c2 - a2) : c3/(a2 - b2)
X(163) lies on these lines: 1,293 19,563 31,923 32,849 48,1094 99,825 101,110 109,112 284,909 643,1018 692,906 798,1101 813,827
X(163) = crosssum of X(656) and X(661)
X(163) = X(I)-aleph conjugate of X(J) for these (I,J): (648,19), (662,610)
X(164) INCENTER OF EXCENTRAL TRIANGLE
Trilinears sin B/2 + sin C/2 - sin A/2 : sin C/2 + sin A/2 - sin B/2 : sin A/2 + sin B/2 - sin C/2Barycentrics a(sin B/2 + sin C/2 - sin A/2) : b(sin C/2 + sin A/2 - sin B/2) : c(sin A/2 + sin B/2 - sin C/2)
X(164) = X(1)-of-excentral triangle
X(164) lies on these lines: 1,258 9,168 40,188 57,177 165,167 173,504 361,503 362,845
X(164) = isogonal conjugate of X(505)
X(164) = X(188)-Ceva conjugate of X(1)
X(164) = X(I)-aleph conjugate of X(J) for these (I,J): (1,361), (2,362), (9,844), (188,164), (366,173)
X(165) = CENTROID OF THE EXCENTRAL TRIANGLE
Trilinears tan(B/2) + tan(C/2) - tan(A/2) : tan(C/2) + tan(A/2) - tan(B/2) : tan(A/2) + tan(B/2) - tan(C/2)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a2 - 2a(b + c) - (b - c)2
Barycentrics a[tan(B/2) + tan(C/2) - tan(A/2)] : b[tan(C/2) + tan(A/2) - tan(B/2)] : c[tan(A/2) + tan(B/2) - tan(C/2)]
X(165) = centroid of the triangle with vertices X(1), X(8), X(20)
X(165) = centroid of the triangle with vertices X(4), X(20), X(40)
X(165) lies on these lines:
1,3 2,516 4,1698 9,910 10,20 32,1571 42,991 43,573 63,100 71,610 105,1054 108,1767 109,212 164,167 166,168 191,1079 210,971 218,1190 220,1615 227,1394 255,1103 269,1253 355,550 371,1703 372,1702 376,515 380,579 386,1695 411,936 479,1323 498,1770 572,1051 574,1572 580,601 612,990 614,902 631,946 750,968 846,1719 950,1788 958,1706 962,11251011,1730 1342,1701 1343,1700
X(165) is the {X(3),X(40)}-harmonic conjugate of X(1). For a list of harmonic conjugates of X(165), click More at the top of this page.
X(165) = isogonal conjugate of X(3062)
X(165) = X(9)-Ceva conjugate of X(1)
X(165) = X(I)-aleph conjugate of X(J) for these (I,J):
(2,169), (9,165), (21,572), (100,101), (188,9), (259,43), (365,978), (366,57), (650,1053)
X(165) = X(I)-beth conjugate of X(J) for these (I,J): (100,165), (643,200)
X(166) GERGONNE POINT OF EXCENTRAL TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (tan A/2)/(cos B/2 + cos C/2 - cos A/2) - (tan B/2)/(cos C/2 + cos A/2 - cos B/2) - (tan C/2)/(cos A/2 + cos B/2 - cos C/2)
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)
X(166) = X(7)-of-excentral triangle
X(166) lies on these lines: 1,1488 165,168 167,188
X(166) = X(266)-cross conjugate of X(57)
X(166) = cevapoint of X(266) and X(289)
X(167) NAGEL POINT OF EXCENTRAL TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = s(B,C,A)t(B,C,A) + s(C,A,B)t(C,A,B) - s(A,B,C)t(A,B,C),
where s(A,B,C) = sin(A/2) and t(A,B,C) = (cos B/2 + cos C/2 - cos A/2) sec A/2
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)
X(167) = X(8)-of-excentral triangle
X(167) lies on these lines: 1,174 164,165 166,188
X(167) = X(9)-aleph conjugate of X(166)
X(168) MITTENPUNKT OF EXCENTRAL TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = sin A - sin B - sin C + 2[cos A/2 + sin(B/2 - A/2) + sin(C/2 - A/2)]
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)
X(168) = X(9)-of-excentral triangle
X(168) lies on these lines: 1,173 9,164 165,166
X(168) = X(188)-aleph conjugate of X(363)
X(169) = X(85)-CEVA CONJUGATE OF X(1)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = - (sin A)cos2(A/2) + (sin B)cos2(B/2) + (sin C)cos2(C/2)
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)
X(169) = X(32)-of-excentral triangle.
X(169) lies on these lines: 1,41 3,910 4,9 6,942 46,672 57,277 63,379 65,218 220,517 572,610
X(169) = X(85)-Ceva conjugate of X(1)
X(169) = crosssum of X(6) and X(1473)
X(169) = X(I)-aleph conjugate of X(J) for these (I,J):
(2,165), (85,169), (86,572), (174,43), (188,170), (508,1), (514,1053), (664,101)
X(170) = X(9)-ALEPH CONJUGATE OF X(9)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = - (tan A/2)sec2(A/2) + (tan B/2)sec2(B/2) + (tan C/2)sec2(C/2)
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)
X(170) = X(76)-of-excentral triangle
X(170) lies on these lines: 1,7 43,218
X(170) = X(220)-Ceva conjugate of X(1)
X(170) = X(I)-aleph conjugate of X(J) for these (I,J): (9,9), (55,43), (188,169), (220,170), (644,1018)
X(170) = X(664)-beth conjugate of X(170)
X(171) = {X(2), X(31)}-HARMONIC CONJUGATE OF X(238)
Trilinears a2 + bc : b2 + ca : c2 + abBarycentrics a3 + abc : b3 + abc : c3 + abc
X(171) lies on these lines: 1,3 2,31 4,601 6,43 7,983 10,58 37,846 42,81 47,498 63,612 72,1046 84,989 98,109 181,511 222,611 292,893 319,757 385,894 388,603 474,978 595,1125 602,631 756,896
X(171) = isogonal conjugate of X(256)
X(171) = X(292)-Ceva conjugate of X(238)
X(171) = X(I)-beth conjugate of X(J) for these (I,J): (100,171), (643,42)
X(172) = TRILINEAR PRODUCT X(6)*X(171)
Trilinears a3 + abc : b3 + abc : c3 + abcBarycentrics a4 + bca2 : b4 + cab2 : c4 + abc2
X(172) lies on these lines:
1,32 6,41 12,230 21,37 35,187 36,39 42,199 58,101 65,248 350,384 577,1038 694,904 699,932
X(172) = isogonal conjugate of X(257)
X(172) = crossdifference of evey pair of points on line X(522)X(1491)
X(172) = X(101)-beth conjugate of X(172)
X(173) = CONGRUENT ISOSCELIZERS POINT
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos B/2 + cos C/2 - cos A/2Trilinears tan(A/2) + sec(A/2) : tan(B/2) + sec(B/2) : tan(C/2) + sec(C/2) (M. Iliev, 4/12/07)
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)
Let P(B)Q(C) be an isoscelizer: let P(B) on sideline AC and Q(C) on AB be equidistant from A, so that AP(B)Q(C) is an isosceles triangle. Line P(B)-to-Q(C), P(C)-to-Q(A), P(A)-to-Q(B) concur in X(173). (P. Yff, unpublished notes, 1989)
The intouch triangle of the intouch triangle of triangle ABC is perspective to triangle ABC, and X(173) is the perspector. (Eric Danneels, Hyacinthos 7892, 9/13/03)
Also, X(173) = X(1486)-of-the-intouch-triangle. (Darij Grinberg; see notes at X(1485) and X(1486).)
If you have The Geometer's Sketchpad, you can view Congruent Isoscelizers Point.
X(173) lies on these lines: 1,168 9,177 57,174 164,504 180,483 503,844 505,1130
X(173) = isogonal conjugate of X(258)
X(173) = X(174)-Ceva conjugate of X(1)
X(173) = X(I)-aleph conjugate of X(J) for these (I,J):
(1,503), (2,504), (174,173), (188,845), (366,164), (507,1), (508,362), (509,361)
X(174) = YFF CENTER OF CONGRUENCE
Trilinears sec A/2 : sec B/2 : sec C/2= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [bc/(b + c - a)]1/2
Barycentrics sin A/2 : sin B/2 : sin C/2
In notes dated 1987, Yff raises a question concerning certain triangles lying within ABC: can three isoscelizers (as defined in connection with X(173), P(B)Q(C), P(C)Q(A), P(A)Q(B) be constructed so that the four triangles P(A)Q(A)A, P(B)Q(B)B, P(C)Q(C)C, ABC are congruent? After proving that the answer is yes, Yff moves the three isoscelizers in such a way that the three outer triangles, P(A)Q(A)A, P(B)Q(B)B, P(C)Q(C)C stay congruent and the inner triangle, ABC, shrinks to X(174).
If you have The Geometer's Sketchpad, you can view Yff Center of Congruence (1) and Yff Center of Congruence (2).
X(174) lies on these lines: 1,167 2,236 7,234 57,173 175,483 176,1143 188,266 481,1127 558,1489
X(174) = isogonal conjugate of X(259)
X(174) = anticomplement of X(2090)
X(174) = X(508)-Ceva conjugate of X(188)
X(174) = cevapoint of X(I) and X(J) for these (I,J): (1,173), (259,266)
X(174) = X(I)-cross conjugate of X(J) for these (I,J): (1,1488), (177,7), (259,188)
X(174) = crosssum of X(1) and X(503)
X(174) = X(556)-beth conjugate of X(556)
X(175) = ISOPERIMETRIC POINT
Trilinears -1 + sec A/2 cos B/2 cos C/2 : -1 + sec B/2 cos C/2 cos A/2 : -1 + sec C/2 cos A/2 cos B/2Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)(-1 + sec A/2 cos B/2 cos C/2)
If a + b + c > 4R + r, where R and r denote the circumradius and inradius, respectively, then there exists a point X for which the perimeters of triangles XBC, XCA, XAB are equal. Veldkamp proved that X = X(175), and Yff, in unpublished notes, proved that X(175) is the center of the outer Soddy circle. See also the 1st and 2nd Eppstein points, X(481), X(482).
The points X(175) and X(176) are discussed in an 1890 article by Emile Lemoine, accessible at Gallica. The article begins on page 111, and the two points are considered beginning on page 128.
Clark Kimberling and R. W. Wagner, Problem E 3020 and Solution, American Mathematical Monthly 93 (1986) 650-652 [proposed 1983].
G. R. Veldkamp, "The isoperimetric point and the point(s) of equal detour," American Mathematical Monthly 92 (1985) 546-558.
X(175) lies on these lines: 1,7 8,1270 174,483 226,1131 490,664 651,1335
X(175) = X(8)-Ceva conjugate of X(176)
X(175) = X(664)-beth conjugate of X(175)
X(175) = {X(1),X(7)}-harmonic conjugate of X(176)
X(176) = EQUAL DETOUR POINT
Trilinears 1 + sec A/2 cos B/2 cos C/2 : 1 + sec B/2 cos C/2 cos A/2 : 1 + sec C/2 cos A/2 cos B/2Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)(1 + sec A/2 cos B/2 cos C/2)
If X is a point not between A and B, we make a detour of magnitude |AX| + |XB| - |AB| if we walk from A to B via X; then a point has the equal detour property if the magnitudes of the three detours, A to B via X, B to C via X, and C to A via X, are equal; X(176) is the only such point unless ABC has an angle greater than 2*arcsin(4/5), and then X(175) also has the equal detour property. Yff found that X(176) is also is the center of the inner Soddy circle. The following construction was found by Elkies: call two circles within ABC companion circles if they are the incircles of two triangles formed by dividing ABC into two smaller triangles by passing a line through one of the vertices and some point on the opposite side; chain of circles O(1), O(2), ... such that O(n),O(n+1) are companion incircles for every n consists of at most six distinct circles; there is a unique chain consisting of only three distinct circles; and for this chain, the three subdividing lines concur in X(176).
The points X(175) and X(176) are discussed in an 1890 article by Emile Lemoine, accessible at Gallica. The article begins on page 111, and the two points are considered beginning on page 128.
G. R. Veldkamp, "The isoperimetric point and the point(s) of equal detour," American Mathematical Monthly 92 (1985) 546-558.
Noam D. Elkies and Jiro Fukuta, Problem E 3236 and Solution, American Mathematical Monthly 97 (1990) 529-531 [proposed 1987].
X(176) lies on these lines: 1,7 8,1271 174,1143 226,1132 489,664 651,1124
X(176) = X(8)-Ceva conjugate of X(175)
X(176) = X(664)-beth conjugate of X(176)
X(176) = {X(1),X(7)}-harmonic conjugate of X(175)
X(177) = 1st MID-ARC POINT
Trilinears (cos B/2 + cos C/2) sec A/2 : (cos C/2 + cos A/2) sec B/2 : (cos A/2 + cos B/2) sec C/2Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)(cos B/2 + cos C/2) sec A/2
Let A',B',C' be the first points of intersection of the angle bisectors of ABC with its incircle. The tangents at A', B', C' form a triangle A"B"C", and the lines AA",BB",CC" concur in X(177). Also, X(177) = X(1) of the intouch triangle.
Clark Kimberling and G. R. Veldkamp, Problem 1160 and Solution, Crux Mathematicorum 13 (1987) 298-299 [proposed 1986].
X(177) is the perspector of ABC and the Yff central triangle, and X(177) is X(65)-of-the-Yff-central-triangle . (Darij Grinberg, Hyacinthos #7689, 8/25/2003)
If you have The Geometer's Sketchpad, you can view X(177) and Yff Central Triangle.
X(177) lies on these lines: 1,167 7,555 8,556 9,173 57,164
X(177) = isogonal conjugate of X(260)
X(177) = X(7)-Ceva conjugate of X(234)
X(177) = crosspoint of X(7) and X(174)
X(177) = crosssum of X(55) and X(259)
X(178) = 2nd MID-ARC POINT
Trilinears (cos B/2 + cos C/2) csc A : (cos C/2 + cos A/2) csc B : (cos A/2 + cos B/2) csc CBarycentrics cos B/2 + cos C/2 : cos C/2 + cos A/2 : cos A/2 + cos B/2
Let A',B',C' be the first points of intersection of the angle bisectors of ABC with its incircle. Let A",B",C" be the midpoints of segments BC,CA,AB, respectively. The lines A'A",B'B",C'C" concur in X(178).
Clark Kimberling, Problem 804, Nieuw Archikef voor Wiskunde 6 (1988) 170.
X(178) lies on these lines: 2,188 8,236
X(178) = complement of X(188)
X(178) = crosspoint of X(2) and X(508)
X(179) = 1st AJIMA-MALFATTI POINT
Trilinears sec4(A/4) : sec4(B/4) : sec4(C/4)Barycentrics sin A sec4(A/4) : sin B sec4(B/4) : sin C sec4(C/4)
The famous Malfatti Problem is to construct three circles O(A), O(B), O(C) inside ABC such that each is externally tangent to the other two, O(A) is tangent to lines AB and AC, O(B) is tangent to BC and BA, and O(C) is tangent to CA and CB. Let A' = O(B)∩O(C), B' = O(C)∩O(A), C' = O(A)∩O(B). The lines AA',BB',CC' concur in X(179). Trilinears are found in Yff's unpublished notes. See also the Yff-Malfatti Point, X(400), having trilinears csc4(A/4) : csc4(B/4) : csc4(C/4), and the references for historical notes.
H. Fukagawa and D. Pedoe, Japanese Temple Geometry Problems (San Gaku), The Charles Babbage Research Centre, Winnipeg, Canada, 1989.
Michael Goldberg, "On the original Malfatti problem," Mathematics Magazine, 40 (1967) 241-247.
Clark Kimberling and I. G. MacDonald, Problem E 3251 and Solution, American Mathematical Monthly 97 (1990) 612-613.
If you have The Geometer's Sketchpad, you can view X(179).
X(179) lies on this line: 1,1142
X(180) = 2nd AJIMA-MALFATTI POINT
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/t(B,C,A) + 1/t(C,B,A) - 1/t(A,C,B),t(A,B,C) = 1 + 2(sec A/4 cos B/4 cos C/4)2
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)
Let A",B",C" be the excenters of ABC, and let A',B',C' be as in the construction of X(179). The lines A'A",B'B",B'B" concur in X(180). Trilinears are found in Yff's unpublished notes. See X(179).
If you have The Geometer's Sketchpad, you can view X(180) and X(180) External.
X(180) lies on this line: 173,483
X(181) = APOLLONIUS POINT
Trilinears a(b + c)2/(b + c - a) : b(c + a)2/(c + a - b) : c(a + b)2/(a + b - c)= a2cos2(B/2 - C/2) : b2cos2(C/2 - A/2) : c2cos2(A/2 - B/2)
Trilinears h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = [r cos(A/2) + s sin(A/2)]2, s = semiperimeter, r = inradius
Barycentrics a3cos2(B/2 - C/2) : b3cos2(C/2 - A/2) : c3cos2(A/2 - B/2)
Let O(A),O(B),O(C) be the excircles. Apollonius's Problem includes the construction of the circle O tangent to the three excircles and encompassing them. (The circle is called the Apollonius circle.) Let A' = O∩O(A), B'=O∩O(B), C'=O∩O(C). The lines AA',BB',CC' concur in X(181). Yff derived trilinears in 1992.
X(181) is the external center of similitude (or exsimilicenter) of the incircle and Apollonius circle. The internal center is X(1682). (Peter J. C. Moses, 8/22/03)
A proof of the the concurrence of lines AA',BB',CC' follows.
A = exsimilicenter(incircle, A-excircle)
A' = exsimilicenter(A-excircle, Apollonius circle)
Let J = exsimilicenter(incircle, Apollonius circle).
By Monge's theorem, the points A, A', J are collinear. In particular, J lies on line AA', and cyclically, J lies on lines BB' and CC'. Therefore, J = X(181). (Darij Grinberg, Hyacinthos, 7461, 8/10/03)
See also
Clark Kimberling, Shiko Iwata, and Hidetosi Fukagawa, Problem 1091 and Solution, Crux Mathematicorum 13 (1987) 128-129; 217-218. [proposed 1985].
X(181) lies on these lines:
1,970 6,197 8,959 10,12 31,51 42,228 43,57 44,375 55,573 56,386 58,1324 171,511 373,748 553,1463 994,1361 1124,1685 1254,1425 1335,1686 1395,1843 1672,1683 1673,1684 1674,1693 1675,1694 1695,1697
X(181) = isogonal conjugate of X(261)
X(181) = X(872)-cross conjugate of X(1500)
X(181) = crosssum of X(I) and X(J) for these (I,J): (21,333), (86,1444)
X(181) = X(I)-beth conjugate of X(J) for these (I,J): (42,181), (660,181), (756,756)
X(182) = MIDPOINT OF BROCARD DIAMETER
Trilinears cos(A - ω) : cos(B - ω) : cos(C - ω)= cos A + sin A tan ω : cos B + sin B tan ω : cos C + sin C tan 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 sin A cos(A - ω) : sin B cos(B - ω) : sin C cos(C - ω)
X(182) is the midpoint of the Brocard diameter (the segment X(3)-to-X(6)); also the center of the 1st Lemoine circle, and the center of the Brocard circle.
If you have The Geometer's Sketchpad, you can view X(1316), which includes X(182).
X(182) lies on these lines:
1,983 2,98 3,6 4,83 5,206 10,1678 22,51 24,1843 30,597 36,1469 40,1700 54,69 55,613 56,611 111,353 140,141 171,1397 373,1495 474,1437 517,1386 518,1385 524,549 692,1001 727,1293 729,1296 952,996 1676,1677
X(182) is the {X(371),X(372)}-harmonic conjugate of X(39). For a list of other harmonic conjugates of X(182), click More at the top of this page.
X(182) = midpoint of X(3) and X(6)
X(182) = reflection of X(I) in X(J) for these (I,J): (6,575), (141,140), (576,6)
X(182) = isogonal conjugate of X(262)
X(182) = isotomic conjugate of X(327)
X(182) = complement of X(1352)
X(183) = TRILINEAR QUOTIENT X(75)/X(182)
Trilinears b2c2cos(A - ω) : c2a2cos(B - ω) : a2b2cos(C - ω)Barycentrics csc A cos(A - ω) : csc B cos(B - ω) : csc C cos(C - ω)
X(183) lies on these lines:
2,6 3,76 5,315 22,157 25,264 55,350 95,305 187,1003 274,474 316,381 317,427 383,621 538,574 622,1080 668,956
X(183) is the {X(2),X(69)}-harmonic conjugate of X(325). For a list of other harmonic conjugates of X(183), click More at the top of this page.
X(183) = isogonal conjugate of X(263)
X(183) = isotomic conjugate of X(262)
X(183) = X(645)-beth conjugate of X(183)
X(184) = INVERSE OF X(125) IN THE BROCARD CIRCLE
Trilinears a2cos A : b2cos B : c2cos CBarycentrics a3cos A : b3cos B : c3cos C
X(184) is the homothetic center of triangles ABC and A'B'C', the latter defined as follows: let B1 and C1 be the points where the perpendicular bisector BC meets sidelines CA and AB, and cyclically define C2, A2; A3, B3. Then A'B'C' is formed by the perpendicular bisectors of segments B1C1, C2A2, A3B3. (Fred Lang, Hyacinthos #1190)
X(184) is the subject of Hyacinthos messages 5423-5441 (May, 2002). In #5423, Alexei Myakishev notes that X(184) serves as a common vertex of three triangles inside ABC, mutually congruent and similar to ABC. (The triangles can be labeled XBcCb, XCaAc, XAbBa, with Bc and Cb on side BC, Ca and Ac on side CA, and Ab and Ba on side AB.) See
Alexei Myakishev, On the Procircumcenter and Related Points , Forum Geometricorum 3 (2003) 29-34.
In #5435, Paul Yiu cites Fred Lang's construction of X(184) and notes that the three triangles are then easily constructed from X(184). The triangles determine three other triangles with common vertex X(184); in #5437, Nikos Dergiades notes that the vertex angles of these are 4A - π, 4B - π, 4C - π, and that
X(184) = X(63)-of-the-orthic-triangle = X(226)-of-the-tangential-triangle
X(184) = homothetic center of the orthic triangle and the medial triangle of the tangential triangle.
X(184) lies on these lines:
2,98 3,49 4,54 5,156 6,25 22,511 23,576 24,389 26,52 22,511 31,604 32,211 48,212 55,215 157,570 160,571 199,573 205,213 251,263 351,686 381,567 397,463 398,462 418,577 572,1011 647,878
X(184) is the {X(6),X(25)}-harmonic conjugate of X(51). For a list of other harmonic conjugates of X(184), click More at the top of this page.
X(184) = isogonal conjugate of X(264)
X(184) = inverse-in-Brocard-circle of X(125)
X(184) = X(I)-Ceva conjugate of X(J) for these (I,J): (6,32), (54,6), (74,50)
X(184) = X(217)-cross conjugate of X(6)
X(184) = crosspoint of X(3) and X(6)
X(184) = crosssum of X(I) and X(J) for these (I,J): (2,4), (5,324), (6, 157), (92,318), (273,342), (338,523), (339,850), (427,1235), (491,492)
X(184) = crossdifference of evey pair of points on line X(297)X(525)
X(184) = X(32)-Hirst inverse of X(237)
X(184) = X(I)-beth conjugate of X(J) for these (I,J): (212,212), (692,184)
X(185) = NAGEL POINT OF THE ORTHIC TRIANGLE
Trilinears (cos A)[1 - cos A cos(B - C)] : (cos B)[1 - cos B cos(C - A)] : (cos C)[1 - cos C cos(A - B)]Barycentrics (sin 2A)[1 - cos A cos(B - C)] : (sin 2B)[1 - cos B cos(C - A)] : (sin 2C)[1 - cos C cos(A - B)]
X(185) lies on these lines:
1,296 3,49 4,51 5,113 6,64 20,193 25,1498 30,52 39,217 54,74 72,916 287,384 378,578 382,568 411,970 648,1105
X(185) = reflection of X(I) in X(J) for these (I,J): (4,389), (125,974)
X(185) = isogonal conjugate of X(1105)
X(185) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,417), (4,235)
X(185) = crosspoint of X(3) and X(4)
X(185) = crosssum of X(I) and X(J) for these (I,J): (3,4), (25,1249)
X(186) = INVERSE-IN-CIRCUMCIRCLE OF X(4)
Trilinears 4 cos A - sec A : 4 cos B - sec B : 4 cos C - sec C= sin 3A csc 2A : sin 3B csc 2B : sin 3C csc 2C
Barycentrics (sin A)(4 cos A - sec A) : (sin B)(4 cos B - sec B) : (sin C)(4 cos C - sec C)
X(186) lies on these lines: 2,3 54,389 93,252 98,935 107,477 112,187 249,250
X(186) is the {X(3),X(24)}-harmonic conjugate of X(4). For a list of other harmonic conjugates of X(100), click More at the top of this page.
X(186) = reflection of X(I) in X(J) for these (I,J): (4,403), (403,468)
X(186) = isogonal conjugate of X(265)
X(186) = isotomic conjugate of X(328)
X(186) = complement of X(3153)
X(186) = anticomplement of X(2072)
X(186) = inverse-in-circumcircle of X(4)
X(186) = X(340)-Ceva conjugate of X(323)
X(186) = X(50)-cross conjugate of X(323)
X(186) = crosspoint of X(54) and X(74)
X(186) = crosssum of X(I) and X(J) for these (I,J): (5,30), (621,622)
X(186) = crossdifference of evey pair of points on line X(216)X(647)
X(187) = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER)
Trilinears a(2a2 - b2 - c2) : b(2b2 - c2 - a2) : c(2c2 - a2 - b2)=sin A - 3 cos A tan ω : sin B - 3 cos B tan ω : sin C - 3 cos C tan ω (Peter J. C. Moses, 8/22/03)
Barycentrics a2(2a2 - b2 - c2) : b2(2b2 - c2 - a2) : c2(2c2 - a2 - b2)
Let L denote the line having trilinears of X(187) as coefficients. Then L is the line passing through X(2) perpendicular to the Euler line.
If you have The Geometer's Sketchpad, you can view X(1316), which includes X(187).
X(187) lies on these lines:
2,316 3,6 23,111 30,115 35,172 36,1015 74,248 99,385 110,352 112,186 183,1003 237,351 249,323 325,620 395,531 396,530 729,805
X(187) is the {X(3),X(6)}-harmonic conjugate of X(574). For a list of other harmonic conjugates of X(187), click More at the top of this page.
X(187) is the radical trace of the circumcircle and Brocard circle. (Peter J. C. Moses, 8/24/03)
X(187) = midpoint of X(I) and X(J) for these (I,J): (15,16), (99,385)
X(187) = reflection of X(I) in X(J) for these (I,J): (115,230), (316,625), (325,620)
X(187) = isogonal conjugate of X(671)
X(187) = inverse-in-circumcircle of X(6)
X(187) = inverse-in-Brocard-circle of X(574)
X(187) = complement of X(316)
X(187) = anticomplement of X(625)
X(187) = X(111)-Ceva conjugate of X(6)
X(187) = crosspoint of X(I) and X(J) for these (I,J): (2,67), (6,111), (468,524)
X(187) = crosssum of X(I) and X(J) for these (I,J): (2,524), (6,23), (111,895), (115,690)
X(187) = crossdifference of evey pair of points on line X(2)X(523)
X(187) = X(55)-beth conjugate of X(187)
X(188) = 2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE
Trilinears csc A/2 : csc B/2 : csc C/2= [bc(b + c - a)]1/2 : [ca(c + a - b)]1/2 : [ab(a + b - c)]1/2
Barycentrics sin A csc A/2 : sin B csc B/2 : sin C csc C/2
X(188) lies on these lines: 1,361 2,178 9,173 40,164 166,167 174,266
X(188) = isogonal conjugate of X(266)
X(188) = anticomplement of X(178)
X(188) = X(2)-Ceva conjugate of X(236)
X(188) = cevapoint of X(1) and X(164)
X(188) = X(259)-cross conjugate of X(174)
X(188) = crosssum of X(1) and X(361)
X(188) = X(188)-beth conjugate of X(266)
X(189) = CYCLOCEVIAN CONJUGATE OF X(8)
Trilinears bc/(cos B + cos C - cos A - 1) : ca/(cos C + cos A - cos B - 1) : ab/(cos A + cos B - cos C - 1)Barycentrics 1/(cos B + cos C - cos A - 1) : 1/(cos C + cos A - cos B - 1) : 1/(cos A + cos B - cos C - 1)
X(189) lies on these lines: 2,77 7,92 8,20 29,81 69,309 222,281
X(189) = isogonal conjugate of X(198)
X(189) = isotomic conjugate of X(329)
X(189) = cyclocevian conjugate of X(8)
X(189) = anticomplement of X(223)
X(189) = X(309)-Ceva conjugate of X(280)
X(189) = cevapoint of X(84) and X(282)
X(189) = X(I)-cross conjugate of X(J) for these (I,J): (4,7), (57,2), (282,280)
X(190) = YFF PARABOLIC POINT
Trilinears bc/(b - c) : ca/(c - a) : ab/(a - b)Barycentrics 1/(b - c) : 1/(c - a) : 1/(a - b)
In unpublished notes, Yff has studied the parabola tangent to sidelines BC, CA, AB and having focus X(101). If A',B',C' are the respective points of tangency, then the lines AA', BB', CC' concur in X(190).
X(190) lies on these lines:
1,537 2,45 6,192 7,344 8,528 9,75 10,671 37,86 40,341 44,239 63,312 69,144 71,290 72,1043 99,101 100,659 110,835 162,643 191,1089 238,726 320,527 321,333 329,345 350,672 513,660 514,1016 522,666 644,651 646,668 649,889 658,1020 670,799 789,813 872,1045
X(190) = reflection of X(I) in X(J) for these (I,J): (239,44), (335,37), (673,9), (903,2)
X(190) = isogonal conjugate of X(649)
X(190) = isotomic conjugate of X(514)
X(190) = anticomplement of X(1086)
X(190) = X(99)-Ceva conjugate of X(100)
X(190) = cevapoint of X(I) and X(J) for these (I,J): (2,514), (9,522), (37,513), (440,525)
X(190) = X(I)-cross conjugate of X(J) for these (I,J): (513,86), (514,2), (522,75)
X(190) = crosssum of X(512) and X(798)
X(190) = X(I)-aleph conjugate of X(J) for these (I,J): (2,1052), (190,1), (645,411), (668,63), (1016,100)
X(190) = X(I)-beth conjugate of X(J) for these (I,J): (9,292), (190,651), (333,88), (645,190), (646,646), (1016,190)
X(190) = trilinear pole of the line X(1)X(2)
Centers 191- 236
are Ceva conjugates. The P-Ceva conjugate of Q is the perspector
of the cevian triangle of P and the anticevian triangle of Q.
X(191) = X(10)-CEVA CONJUGATE OF X(1)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(bc + ca + ab) + b3 + c3 - a3Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(191) lies on these lines:
1,21 9,46 10,267 30,40 35,72 36,960 109,201 165,1079 190,1089 329,498
X(191) = reflection of X(I) in X(J) for these (I,J): (1,21), (79,442)
X(191) = isotomic conjugate of X(267)
X(191) = X(10)-Ceva conjugate of X(1)
X(191) = crosspoint of X(I) and X(J) for these (I,J): (10,502)
X(191) = crosssum of X(58) and X(501)
X(191) = X(I)-aleph conjugate of X(J) for these (I,J): (2,2), (8,20), (10,191), (37,1045), (188,3), (366,6)
X(191) = X(643)-beth conjugate of X(191)
X(192) = X(1)-CEVA CONJUGATE OF X(2)
(CONGRUENT PARALLELIANS POINT)
Trilinears bc(ca + ab - bc) : ca(ab + bc - ca) : ab(bc + ca - ab)Barycentrics ca + ab - bc : ab + bc - ca : bc + ca - ab
Segments through X(192) parallel to the sidelines with endpoints on the sidelines have equal length. For references as early as 1881, see Hyacinthos message 2929 (Paul Yiu, May 29, 2001). See also
Sabrina Bier, "Equilateral Triangles Intercepted by Oriented Parallelians," Forum Geometricorum 1 (2001) 25-32.
X(192) lies on these lines:
1,87 2,37 6,190 7,335 8,256 9,239 55,385 69,742 144,145 315,746 869,1045
X(192) = reflection of X(I) in X(J) for these (I,J): (8,984), (75,37), (1278,75)
X(192) = isogonal conjugate of X(2162)
X(192) = isotomic conjugate of X(330)
X(192) = complement of X(1278)
X(192) = anticomplement of X(75)
X(192) = X(1)-Ceva conjugate of X(2)
X(192) = crosspoint of X(1) and X(43)
X(192) = crosssum of X(1) and X(87)
X(192) = X(9)-Hirst inverse of X(239)
X(192) = X(646)-beth conjugate of X(192)
X(193) = X(4)-CEVA CONJUGATE OF X(2)
Trilinears (csc A)(cot B + cot C - cot A) : (csc B)(cot C + cot A - cot B) : (csc C)(cot A + cot B - cot C)
Barycentrics cot B + cot C - cot A : cot C + cot A - cot B : cot A + cot B - cot C
= 3a2 - b2 - c2 : 3b2 - c2 - a2 : 3c2 - a2 - b2 (Milorad Stevanovic, 5/12/2003)
X(193) lies on these lines:
2,6 7,239 8,894 20,185 23,159 44,344 66,895 144,145 146,148 253,287 317,393 330,959 371,488 372,487 608,651
X(193) = reflection of X(I) in X(J) for these (I,J): (3,1353), (4,1351), (69,6), (1352,576)
X(193) = isotomic conjugate of X(2996)
X(193) = anticomplement of X(69)
X(193) = anticomplementary conjugate of X(1370)
X(193) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,2), (459,439)
X(193) = crosspoint of X(4) and X(459)
X(193) = X(2)-Hirst inverse of X(230)
X(193) = X(I)-beth conjugate of X(J) for these (I,J): (645,193), (662,608)
X(194) = X(6)-CEVA CONJUGATE OF X(2)
Trilinears bc[a2b2 + a2c2 - b2c2] : ca[b2c2 + b2a2 - c2a2] : ab[c2a2 + c2b2 - a2b2]Barycentrics a2b2 + a2c2 - b2c2 : b2c2 + b2a2 - c2a2 : c2a2 + c2b2 - a2b2
Barycentrics cot2A - csc2A cos 2ω : cot2B - csc2B cos 2ω : cot2C - csc2C cos 2ω (M. Iliev, 5/13/07)
X(194) lies on these lines:
1,87 2,39 3,385 4,147 6,384 8,730 20,185 32,99 63,239 69,695 75,1107 257,986 315,736
X(194) is the {X(39),X(76)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(194), click More at the top of this page.
X(194) = reflection of X(76) in X(39)
X(194) = isogonal conjugate of X(3224)
X(194) = isotomic conjugate of X(2998)
X(194) = anticomplement of X(76)
X(194) = anticomplementary conjugate of X(315)
X(194) = eigencenter of cevian triangle of X(6)
X(194) = eigencenter of anticevian triangle of X(2)
X(194) = X(6)-Ceva conjugate of X(2)
X(194) = X(3)-Hirst inverse of X(385)
X(195) = X(5)-CEVA CONJUGATE OF X(3)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(v + w - u),u = u(A,B,C) = cos A cos(B - A) cos(C - A), v = u(B,C,A), w = u(C,A,B)
Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
Barycentrics 4 cos 2A + cot2A - cot A cot ω : 4 cos 2B + cot2B - cot B cot ω : 4 cos 2C + cot2C - cot C cot ω (M. Iliev, 5/13/07)
X(195) lies on the Napoleon cubic and these lines:
1,3467 3,54 4,399 5,3459 6,17 49,52 110,143 140,323 155,381 382,1498 2121,3462 3461,3468
X(195) = reflection of X(I) in X(J) for these (I,J): (3,54), (54,1493), (3519,1209)
X(195) = isogonal conjugate of X(3459)
X(195) = X(5)-Ceva conjugate of X(3)
X(195) = crosssum of X(137) and X(523)
X(196) = X(7)-CEVA CONJUGATE OF X(4)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos B + cos C - cos A - 1) sec A tan A/2Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cos B + cos C - cos A - 1) tan A tan A/2
X(196) lies on these lines:
1,207 2,653 4,65 7,92 19,57 34,937 40,208 55,108 226,281 329,342
X(196) = isogonal conjugate of X(268)
X(196) = X(I)-Ceva conjugate of X(J) for these (I,J): (7,4), (92,278)
X(196) = cevapoint of X(19) and X(207)
X(196) = X(221)-cross conjugate of X(347)
X(196) = X(I)-beth conjugate of X(J) for these (I,J): (648,2) (653,196)
X(197) = X(8)-CEVA CONJUGATE OF X(6)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a[-a2tan A/2 + b2tan B/2 + c2tan C/2]Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
X(197) lies on these lines:
3,10 6,181 19,25 22,100 42,48 56,227 159,200
X(197) = X(8)-Ceva conjugate of X(6)
X(197) = crosssum of X(124) and X(514)
X(198) = X(9)-CEVA CONJUGATE OF X(6)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a(cos B + cos C - cos A - 1)Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
X(198) lies on these lines:
3,9 6,41 19,25 45,1030 64,71 100,346 101,102 154,212 208,227 218,579 284,859 478,577 958,966
X(198) = isogonal conjugate of X(189)
X(198) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,55), (9,6), (223,221)
X(198) = crosspoint of X(40) and X(223)
X(198) = crosssum of X(I) and X(J) for these (I,J): (57,1422), (84,282), (513,1146), (650,1364), (1433,1436)
X(198) = crossdifference of evey pair of points on line X(522)X(905)
X(198) = X(I)-beth conjugate of X(J) for these (I,J): (9,19), (101,198)
X(199) = X(10)-CEVA CONJUGATE OF X(6)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b4 + c4 - a4 + (b2 + c2 - a2)(bc + ca + ab)]Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
X(199) lies on these lines: 2,3 42,172 51,572 55,1030 184,573
X(199) = X(10)-Ceva conjugate of X(6)
X(199) = crosspoint of X(101) and X(250)
X(199) = crosssum of X(125) and X(514)
X(200) = X(8)-CEVA CONJUGATE OF X(9)
Trilinears cot2(A/2) : cot2(B/2) : cot2(C/2)= (b + c - a)2 : (c + a - b) 2 : (a + b - c)2
Barycentrics a(b + c - a)2 : b(c + a - b) 2 : c(a + b - c)2
X(200) lies on these lines:
1,2 3,963 9,55 33,281 40,64 46,1004 57,518 63,100 69,269 159,197 219,282 220,728 255,271 318,1089 319,326 329,516 341,1043 756,968
X(200) is the {X(8),X(78)}-harmonic conjugate of X(1). For a list of harmonic conjugates of X(200), click More at the top of this page.
X(200) = reflection of X(I) in X(J) for these (I,J): (1,997), (57,1376)
X(200) = isogonal conjugate of X(269)
X(200) = isotomic conjugate of X(1088)
X(200) = X(8)-Ceva conjugate of X(9)
X(200) = cevapoint of X(220) and X(480)
X(200) = X(220)-cross conjugate of X(9)
X(200) = crosspoint of X(8) and X(346)
X(200) = crosssum of X(I) and X(J) for these (I,J): (56,1407), (57,1420), (1042,1427)
X(200) = X(I)-beth conjugate of X(J) for these (I,J): (100,223), (200,55), (643,165)
X(201) = X(10)-CEVA CONJUGATE OF X(12)
Trilinears (cos A)[1 + cos(B - C)] : (cos B)[1 + cos(C - A)] : (cos C)[1 + cos(A - B)]Barycentrics (sin 2A)[1 + cos(B - C)] : (sin 2B)[1 + cos(C - A)] : (sin 2C)[1 + cos(A - B)]
X(201) lies on these lines:
1,212 9,34 10,225 12,756 33,40 37,65 38,56 55,774 57,975 63,603 72,73 109,191 210,227 220,221 255,1060 337,348 388,984 601,920
X(201) = isogonal conjugate of X(270)
X(201) = X(10)-Ceva conjugate of X(12)
X(201) = crosspoint of X(10) and X(72)
X(201) = crosssum of X(I) and X(J) for these (I,J): (1,580), (28,58)
X(201) = X(I)-beth conjugate of X(J) for these (I,J): (72,201), (1018,201)
X(202) = X(1)-CEVA CONJUGATE OF X(15)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = u(v + w - u),u = u(A,B,C) = sin(A + π/3), v = u(B,C,A), w = u(C,A,B)
Trilinears 1 - cos(A + π/3) : 1 - cos(B + π/3) : 1 - cos(C + π/3) (Joe Goggins, Oct. 19, 2005)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(202) lies on these lines:
1,62 6,101 11,13 12,18 15,36 16,55 17,499 56,61 395,495 397,496
X(202) = X(1)-Ceva conjugate of X(15)
X(203) = X(1)-CEVA CONJUGATE OF X(16)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = u(v + w - u),u = u(A,B,C) = sin(A - π/3), v = u(B,C,A), w = u(C,A,B)
Trilinears 1 + cos(A + 2π/3) : 1 + cos(B + 2π/3) : 1 + cos(C + 2π/3) (Joe Goggins, Oct. 19, 2005)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(203) lies on these lines:
1,61 6,101 11,14 12,17 15,55 16,36 18,499 56,62 396,495 398,496
X(203) = X(1)-Ceva conjugate of X(16)
X(204) = X(1)-CEVA CONJUGATE OF X(19)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan A)(tan B + tan C - tan A)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(204) lies on these lines: 6,33 19,31 25,34 55,1033 63,162 108,223 207,221
X(204) = X(1)-Ceva conjugate of X(19)
X(204) = X(I)-beth conjugate of X(J) for these (I,J): (108,204), (162,223)
X(205) = X(9)-CEVA CONJUGATE OF X(31)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2[b2tan B/2 + c2tan C/2 - a2tan A/2]Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(205) lies on these lines: 25,41 37,48 78,101 154,220 184,213
X(205) = X(9)-Ceva conjugate of X(31)
X(206) = X(2)-CEVA CONJUGATE OF X(32)
Trilinears a3(b4 + c4 - a4) : b3(c4 + a4 - b4) : c3(a4 + b4 - c4)Barycentrics a4(b4 + c4 - a4) : b4(c4 + a4 - b4) : c4(a4 + b4 - c4)
This is also X(66) of the medial triangle.
X(206) lies on these lines:
2,66 5,182 6,25 26,511 69,110 157,216 160,577 219,692 237,571
X(206) = midpoint of X(I) and X(J) for these (I,J): (6,159), (110,1177)
X(206) = complement of X(66)
X(206) = complementary conjugate of X(427)
X(206) = X(2)-Ceva conjugate of X(32)
X(206) = crosspoint of X(2) and X(315)
X(206) = crosssum of X(339) and X(523)
X(207) = X(1)-CEVA CONJUGATE OF X(34)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 - sec A)(sec B + sec C - sec A - 1)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(207) lies on these lines: 1,196 19,56 33,64 34,1042 40,108 78,653 204,221
X(207) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,34), (196,19)
X(207) = X(1)-beth conjugate of X(64)
X(208) = X(4)-CEVA CONJUGATE OF X(34)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 - sec A)(cos B + cos C - cos A - 1)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(208) lies on these lines:
1,102 4,57 19,225 25,34 33,64 40,196 198,227 226,406 318,653
X(208) = isogonal conjugate of X(271)
X(208) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,34), (57,19), (342,223)
X(208) = crosssum of X(3) and X(1433)
X(208) = X(I)-beth conjugate of X(J) for these (I,J): (108,208), (162,1)
X(209) = X(4)-CEVA CONJUGATE OF X(37)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin B + sin C)[sin A + sin(A - B) + sin(A - C)]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(209) lies on these lines: 6,31 10,12 44,51 306,518
X(209) = isogonal conjugate of X(272)
X(209) = X(4)-Ceva conjugate of X(37)
X(210) = X(10)-CEVA CONJUGATE OF X(37)
Trilinears (b + c)(b + c - a) : (c + a)(c + a - b) : (a + b)(a + b - c)Barycentrics a(b + c)(b + c - a) : b(c + a)(c + a - b) : c(a + b)(a + b - c)
X(210) lies on these lines:
2,354 6,612 8,312 9,55 10,12 31,44 33,220 37,42 38,899 43,984 45,968 51,374 56,936 63,1004 78,958 165,971 201,227 213,762 381,517 392,519 430,594 869,1107 956,997 976,1104
X(210) = X(2)-of-extouch triangle, so that X(210)X(1158) = Euler line of the extouch triangle
X(210) = reflection of X(I) in X(J) for these (I,J): (51,375), (354,2)
X(210) = isogonal conjugate of X(1014)
X(210) = X(10)-Ceva conjugate of X(37)
X(210) = crosspoint of X(8) and X(9)
X(210) = crosssum of X(I) and X(J) for these (I,J): (56,57), (58,1412)
X(210) = crossdifference of evey pair of points on line X(1019)X(1429)
X(210) = X(I)-beth conjugate of X(J) for these (I,J): (200,210), (210,42)
X(211) = X(4)-CEVA CONJUGATE OF X(39)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C)= sin(A + ω)[cos B sin(B + ω) + cos C sin(C + ω) - cos A sin(A + ω)]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(211) lies on these lines: 5,141 32,184 52,114
X(211) = X(4)-Ceva conjugate of X(39)
X(212) = X(9)-CEVA CONJUGATE OF X(41)
Trilinears (cos A)(1 + cos A) : (cos B)(1 + cos B) : (cos C)(1 + cos C)= (cos A)cos2(A/2) : (cos B)cos2(B/2) : (cos C)cos2(C/2)
= a2(b + c - a)(b2 + c2 - a2) : b2(c + a - b)(c2 + a2 - b2) : c2(a + b - c)(a2 + b2 - c2)
Barycentrics (sin 2A)(1 + cos A) : (sin 2B)(1 + cos B) : (sin 2C)(1 + cos C)
X(212) lies on these lines:
1,201 3,73 6,31 9,33 11,748 34,40 35,47 48,184 56,939 63,1040 78,283 109,165 154,198 238,497 312,643 582,942
X(212) = isogonal conjugate of X(273)
X(212) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,48), (9,41), (283,219)
X(212) = X(228)-cross conjugate of X(55)
X(212) = crosspoint of X(I) and X(J) for these (I,J): (3,219), (9,78)
X(212) = crosssum of X(I) and X(J) for these (I,J): (4,278), (34,57)
X(212) = X(212)-beth conjugate of X(184)
X(213) = X(6)-CEVA CONJUGATE OF X(42)
Trilinears (b + c)a2 : (c + a)b2 : (a + b)c2Barycentrics (b + c)a3 : (c + a)b3 : (a + b)c3
X(213) lies on these lines: 1,6 8,981 31,32 39,672 58,101 63,980 83,239 100,729 184,205 274,894 607,1096 667,875 692,923
X(213) = isogonal conjugate of X(274)
X(213) = X(I)-Ceva conjugate of X(J) for these (I,J): (6,42), (37,228)
X(213) = crosspoint of X(6) and X(31)
X(213) = crosssum of X(I) and X(J) for these (I,J): (2,75), (81,1444), (85,348)
X(213) = crossdifference of evey pair of points on line X(320)X(350)
X(213) = X(I)-beth conjugate of X(J) for these (I,J): (41,213), (101,65), (644,213)
X(214) = X(2)-CEVA CONJUGATE OF X(44)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - 2a)(b2 + c2 - a2 - bc)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(214) lies on these lines: 1,88 2,80 9,48 10,140 11,442 36,758 44,1017 119,515 142,528 535,908 662,759 1015,1100
X(214) = midpoint of X(1) and X(100)
X(214) = reflection of X(11) in X(1125)
X(214) = isogonal conjugate of X(1168)
X(214) = complement of X(80)
X(214) = X(2)-Ceva conjugate of X(44)
X(214) = crosspoint of X(2) and X(320)
X(214) = X(21)-beth conjugate of X(244)
X(215) = X(1)-CEVA CONJUGATE OF X(50)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin 3A)(sin 3B + sin 3C - sin 3A)Trilinears cos2(3A/2) : cos2(3B/2) : cos2(3C/2) (M. Iliev, 4/12/07)
Trilinears 1 + cos 3A : 1 + cos 3B : 1+ cos 3C (M. Iliev, 4/12/07)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(215) lies on these lines: 1,49 11,110 12,54 55,184
X(215) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,50)
X(216) = X(5)-CEVA CONJUGATE OF X(51)
Trilinears sin 2A cos(B - C) : sin 2B cos(C - A) : sin 2C cos(A - B)Barycentrics (sin A)(sin 2A)cos(B - C) : sin B sin 2B cos(C - A) : sin C sin 2C cos(A - B)
X(216) lies on these lines:
2,232 3,6 5,53 51,418 95,648 97,288 115,131 157,206 395,465 373,852 395,465 396,466 631,1075 1015,1060
X(216) = isogonal conjugate of X(275)
X(216) = isotomic conjugate of X(276)
X(216) = inverse-in-Brocard-circle of X(577)
X(216) = complement of X(264)
X(216) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,5), (3,418), (5,51), (324,52)
X(216) = cevapoint of X(217) and X(418)
X(216) = X(217)-cross conjugate of X(51)
X(216) = crosspoint of X(I) and X(J) for these (I,J): (2,3), (5,343)
X(216) = crosssum of X(4) and X(6)
X(216) = crossdifference of evey pair of points on line X(186)X(523)
X(217) = X(6)-CEVA CONJUGATE OF X(51)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin3A) cos A cos(B - C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(217) lies on these lines: 4,6 32,184 39,185 54,112 83,287 232,389
X(217) = isogonal conjugate of X(276)
X(217) = X(I)-Ceva conjugate of X(J) for these (I,J): (6,51), (216,418)
X(217) = crosspoint of X(I) and X(J) for these (I,J): (6,184), (51,216)
X(217) = crosssum of X(I) and X(J) for these (I,J): (2,264), (95,275)
X(217) = crossdifference of evey pair of points on line X(340)X(520)
X(218) = X(7)-CEVA CONJUGATE OF X(55)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = cos2(A/2) [cos4(B/2) + cos4(C/2) - cos4(A/2)]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(218) lies on these lines:
1,6 3,41 4,294 7,277 32,906 43,170 46,910 56,101 65,169 145,644 198,579 222,241 279,651
X(218) = isogonal conjugate of X(277)
X(218) = eigencenter of cevian triangle of X(7)
X(218) = eigencenter of anticevian triangle of X(55)
X(218) = X(7)-Ceva conjugate of X(55)
X(218) = crosssum of X(650) and X(1086)
X(218) = X(644)-beth conjugate of X(218)
X(219) = X(8)-CEVA CONJUGATE OF X(55)
Trilinears cos A cot A/2 : cos B cot B/2 : cos C cot C/2= (sin A)/(1 - sec A) : (sin B)/(1 - sec B) : (sin C)/(1 - sec C)
= 1/(csc A - 2 csc 2A) : 1/(csc B - 2 csc 2B) : 1/(csc C - 2 csc 2C)
= a(b + c - a)(b2 + c2 - a2) : b(c + a - b)(c2 + a2 - b2) : c(a + b - c)(a2 + b2 - c2)
Barycentrics sin 2A cot A/2 : sin 2B cot B/2 : sin 2C cot C/2
X(219) lies on these lines:
1,6 3,48 8,29 10,965 19,517 40,610 41,1036 55,284 56,579 63,77 101,102 144,347 200,282 206,692 255,268 278,329 332,345 346,644 572,947 577,906 604,672
X(219) = isogonal conjugate of X(278)
X(219) = isotomic conjugate of X(331)
X(219) = X(I)-Ceva conjugate of X(J) for these (I,J): (8,55), (63,3), (283,212)
X(219) = X(I)-cross conjugate of X(J) for these (I,J): (48,268), (71,9), (212,3)
X(219) = crosspoint of X(I) and X(J) for these (I,J): (8,345), (64,78)
X(219) = crosssum of X(I) and X(J) for these (I,J): (19,34), (56,608)
X(219) = X(I)-beth conjugate of X(J) for these (I,J): (101,478), (219,48), (644,219)
X(220) = X(9)-CEVA CONJUGATE OF X(55)
Trilinears a(b + c - a)2 : b(c + a - b)2 : c(a + b - c)2Trilinears (1 + cos A)2/sin A : (1 + cos B)2/sin B : (1 + cos C)2/sin C (M. Iliev, 4/12/07)
Barycentrics a2(b + c - a)2 : b2(c + a - b)2 : c2(a + b - c)2
X(220) lies on these lines:
1,6 3,101 8,294 33,210 40,910 41,55 48,963 63,241 64,71 78,949 144,279 154,205 169,517 200,728 201,221 268,577 277,1086 281,594 329,948 346,1043
X(220) = isogonal conjugate of X(279)
X(220) = X(I)-Ceva conjugate of X(J) for these (I,J): (9,55), (200,480)
X(220) = cevapoint of X(1) and X(170)
X(220) = crosspoint of X(9) and X(200)
X(220) = crosssum of X(57) and X(269)
X(220) = crossdifference of evey pair of points on line X(513)X(676)
X(220) = X(I)-beth conjugate of X(J) for these (I,J): (101,221), (220,41), (644,220), (728,728)
X(221) = X(1)-CEVA CONJUGATE OF X(56)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin2A/2)(cos B + cos C - cos A - 1)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(221) lies on these lines:
1,84 3,102 6,19 8,651 31,56 40,223 55,64 201,220 204,207 960,1038
X(221) = isogonal conjugate of X(280)
X(221) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,56), (222,6), (223,198)
X(221) = crosspoint of X(I) and X(J) for these (I,J): (1,40), (196,347)
X(221) = crosssum of X(1) and X(84)
X(221) = X(I)-beth conjugate of X(J) for these (I,J): (1,34), (40,40), (101,220), (109,221), (110,3)
X(222) = X(7)-CEVA CONJUGATE OF X(56)
Trilinears cos A tan A/2 : cos B tan B/2 : cos C tan C/2= 1/(csc A + 2 csc 2A) : 1/(csc B + 2 csc 2B) : 1/(csc A + 2 csc 2C)
= a(b2 + c2 - a2)/(b + c - a) : b(c2 + a2 - b2)/(c + a - b) : c(a2 + b2 - c2)/(a + b - c)
Barycentrics a2/(1 + sec A) : b2/(1 + sec B) : c2/(1 + sec C)
X(222) lies on these lines:
1,84 2,651 3,73 6,57 7,27 33,971 34,942 46,227 55,103 56,58 63,77 72,1038 171,611 189,281 218,241 226,478 268,1073 581,1035 601,1066 613,982 912,1060 1355,1363
X(222) = isogonal conjugate of X(281)
X(222) = X(I)-Ceva conjugate of X(J) for these (I,J): (7,56), (77,3), (81,57)
X(222) = cevapoint of X(6) and X(221)
X(222) = X(I)-cross conjugate of X(J) for these (I,J): (48,3), (73,77)
X(222) = crosspoint of X(7) and X(348)
X(222) = crosssum of X(I) and X(J) for these (I,J): (55,607), (650,1146)
X(222) = X(I)-beth conjugate of X(J) for these (I,J):
(21,1012), (63,63), (110,222), (287,222), (648,222), (651,222), (662,2), (895,222)
X(223) = X(2)-CEVA CONJUGATE OF X(57)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan A/2)(cos B + cos C - cos A - 1)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(223) lies on the Thomson cubic and these lines:
1,4 2,77 3,1035 6,57 9,1073 40,221 56,937 63,651 108,204 109,165 312,664 329,347 380,608 580,603 936,1038 1249,3352 3341,3349 3351,3356
X(223) = isogonal conjugate of X(282)
X(223) = complement of X(189)
X(223) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,57), (77,1), (342,208), (347,40)
X(223) = cevapoint of X(198) and X(221)
X(223) = X(I)-cross conjugate of X(J) for these (I,J): (198,40), (227,347)
X(223) = crosspoint of X(2) and X(329)
X(223) = crosssum of X(6) and X(1436)
X(223) = X(I)-aleph conjugate of X(J) for these (I,J):
(63,1079), (77,223), (81,580), (174,46), (651,109)
X(223) = X(I)-beth conjugate of X(J) for these (I,J):
(2,278), (100,200), (162,204), (329,329), (651,223), (662,63)
X(224) = X(7)-CEVA CONJUGATE OF X(63)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = [cot B cos2(B/2) + cot C cos2(C/2) - cot A cos2(A/2)]cot A
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(224) lies on these lines: 1,377 3,63 8,914 21,90 46,100 65,1004 908,1079
X(224) = X(7)-Ceva conjugate of X(63)
X(225) = X(4)-CEVA CONJUGATE OF X(65)
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)
X(225) lies on these lines:
1,4 3,1074 7,969 10,201 12,37 19,208 28,108 46,254 65,407 75,264 91,847 158,1093 377,1038 412,775 653,897
X(225) = isogonal conjugate of X(283)
X(225) = isotomic conjugate of X(332)
X(225) = X(4)-Ceva conjugate of X(65)
X(225) = X(407)-cross conjugate of X(4)
X(225) = crosspoint of X(I) and X(J) for these (I,J): (4,158), (273,278)
X(225) = crosssum of X(I) and X(J) for these (I,J): (3,255), (212,219)
X(225) = X(I)-beth conjugate of X(J) for these (I,J): (4,225), (10,227), (108,1042), (318,10)
X(226) = X(7)-CEVA CONJUGATE OF X(65)
Trilinears (csc A)(cos B + cos C) : (csc B)(cos C + cos A) : (csc C)(cos A + cos B)= bc(b + c)/(b + c - a) : ca(c + a)/(c + a - b) : ab(a + b)/(a + b - c)
Barycentrics (b + c)/(b + c - a) : (c + a)/(c + a - b) : (a + b)/(a + b - c)
This center is also X(63) of the medial triangle.
X(226) lies on these lines:
1,4 2,7 5,912 10,12 11,118 13,1082 14,554 27,284 29,951 35,79 36,1006 37,440 41,379 46,498 55,516 56,405 76,85 78,377 81,651 83,1429 86,1412 92,342 98,109 102,1065 175,1131 176,1132 196,281 208,406 222,478 228,851 262,982 273,469 306,321 429,1426 443,936 452,1420 474,1466 481,485 482,486 495,517 535,551 664,671 673,1174 748,1471 857,1446 975,1038 990,1040 1029,1442 1260,1376 1284,1402 1401,1463
X(226) = reflection of X(993) in X(1125)
X(226) = isogonal conjugate of X(284)
X(226) = isotomic conjugate of X(333)
X(226) = complement of X(63)
X(226) = X(I)-Ceva conjugate of X(J) for these (I,J): (7,65), (349,307)
X(226) = cevapoint of X(37) and X(65)
X(226) = X(I)-cross conjugate of X(J) for these (I,J): (37,10), (73,307)
X(226) = crosspoint of X(2) and X(92)
X(226) = crosssum of X(I) and X(J) for these (I,J): (6,48), (41,55)
X(226) = crossdifference of evey pair of points on line X(652)X(663)
X(226) = X(I)-beth conjugate of X(J) for these (I,J): (2,226), (21,1064), (100,42), (190,226), (312,306), (321,321), (335,226), (835,226)
X(227) = X(10)-CEVA CONJUGATE OF X(65)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(cos B + cos C - cos A - 1)tan A/2Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(227) lies on these lines:
12,37 34,55 40,221 42,65 46,222 56,197 198,208 201,210 322,347 607,910
X(227) = isogonal conjugate of X(285)
X(227) = X(10)-Ceva conjugate of X(65)
X(227) = crosspoint of X(223) and X(347)
X(227) = crosssum of X(84) and X(1433)
X(227) = X(I)-beth conjugate of X(J) for these (I,J): (10,225), (40,227), (100,72)
X(228) = X(3)-CEVA CONJUGATE OF X(71)
Trilinears (sin 2A)(sin B + sin C) : (sin 2B)(sin C + sin A) : (sin 2C)(sin A + sin B)Barycentrics (sin A sin 2A)(sin B + sin C) : (sin B sin 2B)(sin C + sin A) : (sin C sin 2C)(sin A + sin B)
X(228) lies on these lines:
3,63 9,1011 12,407 19,25 28,943 31,32 35,846 42,181 48,184 73,408 98,100 226,851
X(228) = isogonal conjugate of X(286)
X(228) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,71), (37,213), (55,42)
X(228) = crosspoint of X(I) and X(J) for these (I,J): (3,48), (37,72), (55,212), (71,73)
X(228) = crosssum of X(I) and X(J) for these (I,J): (4,92), (7,273), (27,29), (28,81)
X(228) = crossdifference of evey pair of points on line X(693)X(905)
X(228) = X(212)-beth conjugate of X(228)
X(229) = X(7)-CEVA CONJUGATE OF X(81)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (v + w - u)/(b + c),u = u(a,b,c) = a(b + c - a)/(b + c), v = u(b,c,a), w = u(c,a,b)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(229) lies on these lines: 1,267 21,36 28,60 58,244 65,110 593,1104
X(229) = midpoint of X(1) and X(267)
X(229) = X(7)-Ceva conjugate of X(81)
X(230) = X(2)-CEVA CONJUGATE OF X(114)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc[a2(2a2 - b2 - c2) + (b2 - c2)2]
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(230) lies on these lines:
2,6 5,32 12,172 25,53 30,115 39,140 50,858 111,476 112,403 231,232 393,459 427,571 538,620 549,574 625,754
X(230) = midpoint of X(I) and X(J) for these (I,J): (115,187), (325,385), (395,396)
X(230) = isogonal conjugate of X(2987)
X(230) = complement of X(325)
X(230) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,114), (297,1503)
X(230) = crosspoint of X(2) and X(98)
X(230) = crosssum of X(6) and X(511)
X(230) = crossdifference of evey pair of points on line X(3)X(512)
X(230) = X(2)-Hirst inverse of X(193)
X(230) = X(I)-beth conjugate of X(J) for these (I,J): (281,230), (645,230)
X(231) = X(2)-CEVA CONJUGATE OF X(128)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = u(-au + bv + cw), u : v : w = X(128)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(231) lies on these lines: 4,96 6,17 50,115 230,232
X(231) = complement of X(1273)
X(231) = X(2)-Ceva conjugate of X(128)
X(231) = crosssum of X(6) and X(1154)
X(231) = crossdifference of evey pair of points on line X(3)X(1510)
X(231) = X(281)-beth conjugate of X(230)
X(232) = X(2)-CEVA CONJUGATE OF X(132)
Trilinears tan A cos(A + ω) : tan B cos(B + ω) : tan C cos(C + ω)Barycentrics sin A tan A cos(A + ω) : sin B tan B cos(B + ω) : sin C tan C cos(C + ω)
X(232) lies on these lines:
2,216 4,39 6,25 19,444 22,577 23,250 24,32 53,427 112,186 115,403 217,389 230,231 297,325 378,574 385,648 459,800
X(232) = midpoint of X(3269) annd X(3331)
X(232) = isogonal conjugate of X(287)
X(232) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,132), (297,511)
X(232) = X(237)-cross conjugate of X(511)
X(232) = crosssum of X(2) and X(401)
X(232) = crossdifference of evey pair of points on line X(3)X(525)
X(232) = orthojoin of X(132)
X(232) = X(6)-Hirst inverse of X(25)
X(232) = X(281)-beth conjugate of X(232)
X(233) = X(2)-CEVA CONJUGATE OF X(140)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [b cos(C - A) + c cos(B - A)]cos(B - C)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(233) lies on these lines: 2,95 5,53 6,17 115,128 122,138
X(233) = isogonal conjugate of X(288)
X(233) = complement of X(95)
X(233) = X(2)-Ceva conjugate of X(140)
X(233) = crosspoint of X(2) and X(5)
X(233) = crosssum of X(6) and X(54)
X(233) = crossdifference of evey pair of points on line X(1157)X(1510)
X(234) = X(7)-CEVA CONJUGATE OF X(177)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos B/2 + cos C/2)(cos B/2 cos C/2)2Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(234) lies on these lines: 2,178 7,174 57,362 75,556 555,1088
X(234) = X(7)-Ceva conjugate of X(177)
X(235) = X(4)-CEVA CONJUGATE OF X(185)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A - cos(B - C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(235) lies on these lines: 2,3 11,34 12,33 52,113 133,136
X(235) = midpoint of X(4) and X(24)
X(235) = X(4)-Ceva conjugate of X(185)
X(235) = crosssum of X(3) and X(1092)
X(236) = X(2)-CEVA CONJUGATE OF X(188)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A/2)(cos B/2 + cos C/2 - cos A/2)Trilinears [1 + sin(A/2)]/sin A : [1 + sin(B/2)]/sin B : [1 + sin(C/2)]//sin C (M. Iliev, 4/12/07)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(236) lies on these lines: 2,174 8,178 9,173
X(236) = isogonal conjugate of X(289)
X(236) = X(2)-Ceva conjugate of X(188)
Centers 237- 248
are line conjugates. The P-line conjugate of Q is the point
where line PQ meets the trilinear polar of the isogonal conjugate of Q.
X(237) = X(3)-LINE CONJUGATE OF X(2)
Trilinears a2cos(A + ω) : b2cos(B + ω) : c2cos(C + ω)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b4 + c4 - a2b2 - a2c2) (Darij Grinberg, 3/29/03)
Barycentrics a3cos(A + ω) : b3cos(B + ω) : c3cos(C + ω)
X(237) is the point of intersection of the Euler line and the Lemoine axis (defined as the radical axis of the circumcircle and the Brocard circle).
If you have The Geometer's Sketchpad, you can view X(1316), which includes X(237).
X(237) lies on these lines: 2,3 6,160 31,904 32,184 39,51 154,682 187,351 206,571
X(237) is the {X(1113),X(1114)}-harmonic conjugate of X(1316). For a list of other harmonic conjugates of X(237), click More at the top of this page.
X(237) = isogonal conjugate of X(290)
X(237) = X(98)-Ceva conjugate of X(6)
X(237) = crosspoint of X(I) and X(J) for these (I,J): (6,98), (232,511)
X(237) = crosssum of X(I) and X(J) for these (I,J): (2,511), (98,287)
X(237) = crossdifference of evey pair of points on line X(2)X(647)
X(237) = X(32)-Hirst inverse of X(184)
X(237) = X(3)-line conjugate of X(2)
X(237) = X(55)-beth conjugate of X(237)
X(238) = X(1)-LINE CONJUGATE OF X(37)
Trilinears a2 - bc : b2 - ca : c2 - abBarycentrics a3 - abc : b3 - abc : c3 - abc
X(238) lies on these lines:
1,6 2,31 3,978 4,602 7,1471 8,983 10,82 21,256 36,513 40,1722 42,1621 43,55 47,499 56,87 57,1707 58,86 63,614 71,1244 100,899 105,291 106,898 162,415 190,726 212,497 239,740 241,1456 242,419 244,896 390,1253 459,1395 484,1739 516,673 517,1052 519,765 580,946 601,631 651,1458 662,1326 942,1046 987,1472 992,1009 993,995 1006,1064 1040,1711 1054,1155 1284,1428 1465,1758 1479,1714 1699,1754
X(238) = midpoint of X(1) and X(1279)
X(238) = reflection of X(1) in X(1297)
X(238) = isogonal conjugate of X(291)
X(238) = isotomic conjugate of X(334)
X(238) = X(I)-Ceva conjugate of X(J) for these (I,J): (105,1), (292,171)
X(238) = crosssum of X(I) and X(J) for these (I,J): (10,726), (42,672), (239,894)
X(238) = crossdifference of evey pair of points on line X(37)X(513)
X(238) = X(I)-Hirst inverse of X(J) for these (I,J): (1,6), (43,55)
X(238) = X(1)-line conjugate of X(37)
X(238) = X(105)-aleph conjugate of X(238)
X(238) = X(I)-beth conjugate of X(J) for these (I,J): (21,238), (643,902), (644,238), (932,238)
X(239) = X(1)-LINE CONJUGATE OF X(42)
Trilinears bc(a2 - bc) : ca(b2 - ca) : ab(c2 - ab)Barycentrics a2 - bc : b2 - ca : c2 - ab
X(239) lies on these lines:
1,2 6,75 7,193 9,192 44,190 57,330 63,194 81,274 83,213 86,1100 92,607 141,319 238,740 241,664 257,333 294,666 318,458 320,524 335,518 514,649 1043,1104
X(239) = reflection of X(I) in X(J) for these (I,J): (190,44), (320,1086)
X(239) = isogonal conjugate of X(292)
X(239) = isotomic conjugate of X(335)
X(239) = crosspoint of X(256) and X(291)
X(239) = crosssum of X(I) and X(J) for these (I,J): (3,255), (212,219)
X(239) = crossdifference of evey pair of points on line X(42)X(649)
X(239) = X(I)-Hirst inverse of X(J) for these (I,J): (171,238), (665,1015)
X(239) = X(1)-line conjugate of X(42)
X(239) = X(I)-beth conjugate of X(J) for these (I,J): (333,239), (645,44)
X(240) = X(1)-LINE CONJUGATE OF X(48)
Trilinears sec A cos(A + ω) : sec B cos(B + ω) : sec C cos(C + ω)Barycentrics tan A cos(A + ω) : tan B cos(B + ω) : tan C cos(C + ω)
X(240) lies on these lines: 1,19 4,256 38,92 63,1096 75,158 162,896 278,982 281,984 522,656 607,611 608,613
X(240) = isogonal conjugate of X(293)
X(240) = isotomic conjugate of X(336)
X(240) = crossdifference of evey pair of points on line X(48)X(656)
X(240) = X(1)-Hirst inverse of X(19)
X(240) = X(1)-line conjugate of X(48)
X(240) = X(318)-beth conjugate of X(240)
X(241) = X(1)-LINE CONJUGATE OF X(55)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = cos4B/2 - [cos2(A/2)][cos2(B/2) +cos2(C/2)] + cos4(C/2)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(241) lies on these lines: 1,3 2,85 6,77 7,37 9,269 44,651 63,220 141,307 218,222 239,664 277,278 294,910 347,1108 514,650 960,1042
X(241) = isogonal conjugate of X(294)
X(241) = crosssum of X(I) and X(J) for these (I,J): (6,910), (518,1376
X(241) = crossdifference of evey pair of points on line X(55)X(650)
X(241) = X(1)-Hirst inverse of X(57)
X(241) = X(1)-line conjugate of X(55)
X(241) = X(I)-beth conjugate of X(J) for these (I,J): (2,241), (100,241), (1025,241), (1026,241)
X(242) = X(4)-LINE CONJUGATE OF X(71)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)(sin2A - sin B sin C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(242) lies on these lines: 4,9 25,92 28,261 29,257 34,87 162,422 238,419 278,459 915,929
X(242) = isogonal conjugate of X(295)
X(242) = isotomic conjugate of X(337)
X(242) = crossdifference of evey pair of points on line X(71)X(1459)
X(242) = X(4)-Hirst inverse of X(19)
X(242) = X(4)-line conjugate of X(71)
X(243) = X(4)-LINE CONJUGATE OF X(73)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)(cos2A - cos B cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(243) lies on these lines: 1,4 3,158 55,92 65,412 318,958 411,821 425,662 522,652 920,1075 1040,1096
X(243) = isogonal conjugate of X(296)
X(243) = crossdifference of evey pair of points on line X(73)X(652)
X(243) = X(I)-Hirst inverse of X(J) for these (I,J): (1,4), (46,1148)
X(243) = X(1)-line conjugate of X(73)
X(244) = X(1)-LINE CONJUGATE OF X(100)
Trilinears (b - c)2 : (c - a)2 : (a - b)2= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [1 - cos(B - C)]sin2(A/2)
Barycentrics a(b - c)2 : b(c - a)2 : c(a - b)2
X(244) lies on these lines: 1,88 2,38 11,867 31,57 34,1106 42,354 58,229 63,748 238,896 474,976 518,899 596,1089 665,866
X(244) = isogonal conjugate of X(765)
X(244) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,513), (75,514)
X(244) = crosspoint of X(1) and X(513)
X(244) = crosssum of X(I) and X(J) for these (I,J): (1,100), (31,101), (78,1331), (109,1420), (200,644), (651,1445), (678,1023), (756,1018)
X(244) = crossdifference of evey pair of points on line X(100)X(101)
X(244) = X(1)-Hirst inverse of X(1054)
X(244) = X(1)-line conjugate of X(100)
X(245) = X(1)-LINE CONJUGATE OF X(110)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = csc2(C - A) + csc(C - B) [csc(C - A) -csc(B - A)] + csc2(A - B)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(245) lies on these lines: 1,60 115,125
X(245) = X(1)-line conjugate of X(110)
X(246) = X(3)-LINE CONJUGATE OF X(110)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = csc(B-A)[cos A csc(B - A) + cos C csc(B - C)] + csc(C - A) u(A,B,C),
u(A,B,C) = [cos A csc(C - A) + cos B csc(C - B)]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(246) lies on these lines: 3,74 115,125
X(246) = X(3)-line conjugate of X(110)
X(247) = X(4)-LINE CONJUGATE OF X(110)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = csc(B-A)[sec A csc(B - A) + sec C csc(B - C)] + csc(C - A) u(A,B,C),
u(A,B,C) = [sec A csc(C - A) + sec B csc(C - B)]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(247) lies on these lines: 4,110 115,125
X(247) = crossdifference of evey pair of points on line X(110)X(686)
X(247) = X(4)-line conjugate of X(110)
X(248) = X(4)-LINE CONJUGATE OF X(132)
Trilinears sin 2A sec(A + ω) : sin 2B sec(B + ω) : sin 2C sec(C + ω)Barycentrics sin A sin 2A sec(A + ω) : sin B sin 2B sec(B + ω) : sin C sin 2C sec(C + ω)
X(248) lies on these lines:
4,32 6,157 39,54 50,67 65,172 66,571 69,287 72,293 74,187 290,385 682,695
X(248) = isogonal conjugate of X(297)
X(248) = crosspoint of X(98) and X(287)
X(248) = crosssum of X(232) and X(511)
X(248) = crossdifference of evey pair of points on line X(114)X(132)
X(248) = X(4)-line conjugate of X(132)
Centers 249- 297
are isogonal conjugates of previously listed centers.
X(249) = ISOGONAL CONJUGATE OF X(115)
Trilinears (csc A)csc2(B - C) : (csc B)csc2(C - A) : (csc C)csc2(A - B)= a/(b2 - c2)2 : b/(c2 - a2)2 : c/(a2 - b2)2
Barycentrics csc2(B - C) : csc2(C - A) : csc2(A - B)
X(249) lies on these lines: 99,525 110,512 186,250 187,323 297,316 648,687 805,827 849,1110
X(249) = isogonal conjugate of X(115)
X(249) = isotomic conjugate of X(338)
X(249) = cevapoint of X(I) and X(J) for these (I,J): (6,110), (24,112)
X(249) = X(I)-cross conjugate of X(J) for these (I,J): (3,99), (6,110)
X(250) = ISOGONAL CONJUGATE OF X(125)
Trilinears (sec A)csc2(B - C) : (sec B)csc2(C - A) : (sec C)csc2(A - B)= (a2sec A)/(b2 - c2)2 : (b2sec B)/(c2 - a2)2 : (c2sec C)/(a2 - b2)2
Barycentrics (tan A)csc2(B - C) : (tan B)csc2(C - A) : (tan C)csc2(A - B)
X(250) lies on these lines: 23,232 107,687 110,520 112,691 186,249 325,340 476,933 523,648 827,935
X(250) = isogonal conjugate of X(125)
X(250) = isotomic conjugate of X(339)
X(250) = cevapoint of X(I) and X(J) for these (I,J): (3,110), (25,112), (162,270)
X(250) = X(I)-cross conjugate of X(J) for these (I,J): (3,110), (22,99), (24,107), (25,112), (199,101)
X(251) = ISOGONAL CONJUGATE OF X(141)
Trilinears a2csc(A + ω) : b2csc(B + ω) : c2csc(C + ω)= a/(b2 + c2) : b/(c2 + a2) : c/(a2 + b2)
Barycentrics a3csc(A + ω) : b3csc(B + ω) : c3csc(C + ω)
X(251) lies on these lines: 2,32 6,22 37,82 110,694 112,427 184,263 308,385 609,614 689,699
X(251) = isogonal conjugate of X(141)
X(251) = complement of X(1369)
X(251) = cevapoint of X(6) and X(32)
X(251) = X(I)-cross conjugate of X(J) for these (I,J): (6,83), (23,111), (523,112)
X(252) = ISOGONAL CONJUGATE OF X(143)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = sec(B - C)/[1 - 2 cos(2A)]
Trilinears h(A,B,C) : h(B,C,A) : h(C,A,B), where
h(A,B,C) = cos A sec(3A) sec(B - C) (Manol Iliev, 4/01/07)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C) f(C,A,B)
X(252) lies on these lines: 3,930 54,140 93,186
X(252) = isogonal conjugate of X(143)
X(253) = X(4)-CROSS CONJUGATE OF X(2)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)/(tan B + tan C - tan A)= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (csc2A)/(cos A - cos B cos C)
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/(tan B + tan C - tan A)
X(253) lies on these lines: 2,1073 7,280 8,307 20,64 193,287 306,329 318,342 322,341
X(253) = isogonal conjugate of X(154)
X(253) = isotomic conjugate of X(20)
X(253) = cyclocevian conjugate of X(69)
X(253) = cevapoint of X(122) and X(525)
X(253) = X(I)-cross conjugate of X(J) for these (I,J): (4,2), (122,525)
X(254) = X(3)-CROSS CONJUGATE OF X(4)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)/(cos2B + cos2C - cos2A)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (tan A)/(cos2B + cos2C - cos2A)
X(254) lies on these lines: 2,847 4,155 24,393 46,225 68,136
X(254) = isogonal conjugate of X(155)
X(254) = cevapoint of X(136) and X(523)
X(254) = X(3)-cross conjugate of X(4)
X(255) = ISOGONAL CONJUGATE OF X(158)
Trilinears cos2A : cos2B : cos2C= 1 + cos 2A : 1 + cos 2B : 1 + cos 2C
Barycentrics sin A cos2A : sin B cos2B : sin C cos2C
X(255) lies on these lines: 1,21 3,73 35,991 36,1106 40,109 48,563 55,601 56,602 57,580 91,1109 92,1087 158,775 162,1099 165,1103 200,271 201,1060 219,268 293,304 326,1102 411,651 498,750 499,748
X(255) = isogonal conjugate of X(158)
X(255) = X(I)-Ceva conjugate of X(J) for these (I,J): (63,48), (283,3)
X(255) = crosspoint of X(63) and X(326)
X(255) = crosssum of X(I) and X(J) for these (I,J): (1,290), (4,1068), (19,1096)
X(255) = X(I)-aleph conjugate of X(J) for these (I,J): (775,255), (1105,158)
X(256) = 1st SHARYGIN POINT
Trilinears 1/(a2 + bc) : 1/(b2 + ca) : 1/(c2 + ab)Barycentrics a/(a2 + bc) : b/(b2 + ca) : c/(c2 + ab)
See the description at X(1281). The lines AD, BE, CF defined there concur in X(256).
X(256) lies on these lines: 1,511 3,987 4,240 7,982 8,192 9,43 21,238 37,694 40,989 55,983 84,988 104,1064 291,894 314,350 573,981
X(256) = isogonal conjugate of X(171)
X(256) = isotomic conjugate of X(1909)
X(256) = X(239)-cross conjugate of X(291)
X(256) = crosssum of X(43) and X(846)
X(256) = X(238)-Hirst inverse of X(904)
X(257) = ISOGONAL CONJUGATE OF X(172)
Trilinears 1/(a3 + abc) : 1/(b3 + abc) : 1/(c3 + abc)Barycentrics 1/(a2 + bc) : 1/(b2 + ca) : 1/(c2 + ab)
X(257) lies on these lines: 1,385 8,192 29,242 65,894 75,698 92,297 194,986 239,333 330,982 335,694
X(257) = isogonal conjugate of X(172)
X(527) = isotomic conjugate of X(894)
X(257) = X(350)-cross conjugate of X(335)
X(257) = X(239)-Hirst inverse of X(893)
X(258) = CONGRUENT INCIRCLES ISOSCELIZER POINT
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(cos B/2 + cos C/2 - cos A/2)= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1 + sin(B/2) + sin(C/2) - sin(A/2)
Trilinears tan(A/2) - sec(A/2) : tan(B/2) - sec(B/2) : tan(C/2) - sec(C/2) (M. Iliev, 4/12/07)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
In Yff's isoscelizer configuration, if X = X(258), then the isosceles triangles Ta, Tb, Tc have congruent incircles.
If you have The Geometer's Sketchpad, you can view X(258).
X(258) lies on these lines: 1,164 57,173 259,289
X(258) = isogonal conjugate of X(173)
X(258) = X(259)-cross conjugate of X(1)
X(258) = X(366)-aleph conjugate of X(363)
X(259) = ISOGONAL CONJUGATE OF X(174)
Trilinears cos A/2 : cos B/2 : cos C/2= [a(b + c - a)]1/2 : [b(c + a - b)]1/2 : [c(a + b - c)]1/2
Barycentrics sin A cos A/2 : sin B cos B/2 : sin C cos C/2
X(259) lies on these lines: 1,168 258,289 260,266
X(259) = isogonal conjugate of X(174)
X(259) = X(I)-Ceva conjugate of X(J) for these (I,J): (174,266), (260,55)
X(259) = cevapoint of X(1) and X(503)
X(259) = crosspoint of X(I) and X(J) for these (I,J): (1,258), (174,188)
X(259) = crosssum of X(I) and X(J) for these (I,J): (1,173), (259,266)
X(260) = ISOGONAL CONJUGATE OF X(177)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A/2)/(cos B/2 + cos C/2)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(260) lies on these lines: 1,3 259,266
X(260) = isogonal conjugate of X(177)
X(260) = cevapoint of X(55) and X(259)
X(261) = ISOTOMIC CONJUGATE OF X(12)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [(csc A)(sec(B/2 - C/2))]2Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(261) lies on these lines:
2,593 9,645 21,314 28,242 58,86 75,99 272,310 284,332 317,406 319,502 552,873 572,662
X(261) = isogonal conjugate of X(181)
X(261) = isotomic conjugate of X(12)
X(261) = X(873)-Ceva conjugate of X(1509)
X(261) = cevapoint of X(21) and X(333)
X(262) = ISOGONAL CONJUGATE OF X(182)
Trilinears sec(A - ω) : sec(B - ω) : sec(C - ω)Barycentrics sin A sec(A - ω) : sin B sec(B - ω) : sin C sec(C - ω)
X(262) lies on these lines: 2,51 3,83 4,39 5,76 6,98 13,383 14,1080 25,275 30,598 226,982 381,671 385,576
X(262) = isogonal conjugate of X(182)
X(262) = isotomic conjugate of X(183)
X(263) = ISOGONAL CONJUGATE OF X(183)
Trilinears a2sec(A - ω) : b2sec(B - ω) : c2sec(C - ω)Barycentrics a3sec(A - ω) : b3sec(B - ω) : c3sec(C - ω)
X(263) lies on these lines: 2,51 6,160 69,308 184,251
X(263) = isogonal conjugate of X(183)
X(264) = ISOTOMIC CONJUGATE OF CIRCUMCENTER
Trilinears csc A csc 2A : csc B csc 2B : csc C csc 2C= sec A csc2A : sec B csc2B : sec C csc2C
= tan A csc(A - ω) : tan B csc(B - ω) : tan C csc(C - ω)
Barycentrics csc 2A : csc 2B : csc 2C
X(264) lies on these lines:
2,216 3,95 4,69 5,1093 6,287 25,183 33,350 53,141 75,225 85,309 92,306 99,378 274,475 281,344 298,472 299,473 300,302 301,303 305,325 339,381 379,823 401,577
X(264) = isogonal conjugate of X(184)
X(264) = isotomic conjugate of X(3)
X(264) = complement of X(3164)
X(264) = anticomplement of X(216)
X(264) = X(276)-Ceva conjugate of X(2)
X(264) = cevapoint of X(I) and X(J) for these (I,J): (2,4), (5,324), (6,157), (92,318), (273,342), (338,523), (491,492)
X(264) = X(I)-cross conjugate of X(J) for these (I,J): (2,76), (5,2), (30,94), (92,331), (427,4), (442,321)
X(265) = REFLECTION OF X(3) IN X(125)
Trilinears sin 2A csc 3A : sin 2B csc 3B : sin 2C csc 3C= 1/(4 cos A - sec A) : 1/(4 cos B - sec B) : 1/(4 cos C sec C)
Barycentrics sin A sin 2A csc 3A : sin B sin 2B csc 3B : sin C sin 2C csc 3C
X(265) lies on these lines: 3,125 4,94 5,49 6,13 30,74 64,382 65,79 67,511 69,328 290,316 300,621 301,622
X(265) = midpoint of X(4) and X(3448)
X(265) = reflection of X(I) in X(J) for these (I,J): (3,125), (110,5), (146,1539), (399,113)
X(265) = isogonal conjugate of X(186)
X(265) = isotomic conjugate of X(340)
X(265) = cevapoint of X(5) and X(30)
X(265) = crosspoint of X(94) and X(328)
X(266) = ISOGONAL CONJUGATE OF X(188)
Trilinears sin A/2 : sin B/2 : sin C/2= [a/(b + c - a)]1/2 : [b/(c + a - b)]1/2 : [c/(a + b - c)]1/2
Barycentrics sin A sin A/2 : sin B sin B/2 : sin C sin C/2
X(266) lies on these lines:1,164 56,289 174,188 259,260 361,978
X(266) = isogonal conjugate of X(188)
X(266) = eigencenter of cevian triangle of X(174)
X(266) = eigencenter of anticevian triangle of X(259)
X(266) = X(174)-Ceva conjugate of X(259)
X(266) = cevapoint of X(1) and X(361)
X(266) = X(6)-cross conjugate of X(289)
X(266) = crosspoint of X(1) and X(505)
X(266) = crosssum of X(1) and X(164)
X(267) = ISOGONAL CONJUGATE OF X(191)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = 1/[b3 + c3 - a3 + (b + c - a)(bc + ca + ab)]
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(267) lies on these lines: 1,229 10,191 35,37
X(267) = reflection of X(1) in X(229)
X(267) = isogonal conjugate of X(191)
X(267) = cevapoint of X(58) and X(501)
X(267) = X(58)-cross conjugate of X(1)
X(268) = ISOGONAL CONJUGATE OF X(196)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A cot A/2)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(268) lies on these lines: 3,9 21,280 219,255 220,577 222,1073 281,1012
X(268) = isogonal conjugate of X(196)
X(268) = X(I)-cross conjugate of X(J) for these (I,J): (48,219), (55,3)
X(268) = crosssum of X(19) and X(207)
X(269) = ISOGONAL CONJUGATE OF X(200)
Trilinears tan2A/2 : tan2B/2 : tan2C/2= [a2 - (b - c)2]2 : [b2 - (c - a)2]2 : [c2 - (a - b)2]2 Barycentrics sin A tan2A/2 : sin B tan2B/2 : sin C tan2C/2
X(269) lies on these lines: 1,7 3,939 6,57 9,241 46,1103 56,738 69,200 86,1088 106,934 142,948 273,1111 292,1020 307,936 320,326 479,614
X(269) = isogonal conjugate of X(200)
X(269) = isotomic conjugate of X(341)
X(269) = X(279)-Ceva conjugate of X(57)
X(269) = X(56)-cross conjugate of X(57)
X(269) = crosspoint of X(279) and X(479)
X(269) = crosssum of X(220) and X(480)
X(270) = ISOGONAL CONJUGATE OF X(201)
Trilinears (sec A)/[1 + cos(B - C)] : (sec B)/[1 + cos(C - A)] : (sec C)/[1 + cos(A - B)]Barycentrics (tan A)/[1 + cos(B - C)] : (tan B)/[1 + cos(C - A)] : (tan C)/[1 + cos(A - B)]
X(270) lies on these lines: 4,162 27,58 28,60 29,283 759,933
X(270) = isogonal conjugate of X(201)
X(270) = X(250)-Ceva conjugate of X(162)
X(270) = cevapoint of X(28) and X(58)
X(270) = X(58)-cross conjugate of X(60)
X(271) = ISOGONAL CONJUGATE OF X(208)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cot A cot A/2)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(271) lies on these lines: 2,1034 8,20 78,394 200,255 282,283
X(271) = isogonal conjugate of X(208)
X(271) = isotomic conjugate of X(342)
X(271) = X(I)-cross conjugate of X(J) for these (I,J): (3,78), (9,63)
X(272) = ISOGONAL CONJUGATE OF X(209)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = 1/[(sin A + sin(A - B) + sin(A - C))(sin B + sin C)]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(272) lies on these lines: 2,284 7,58 21,75 28,273 60,86 261,310 1014,1088
X(272) = isogonal conjugate of X(209)
X(272) = X(3)-cross conjugate of X(81)
X(273) = ISOGONAL CONJUGATE OF X(212)
Trilinears sec A sec2(A/2) : sec B sec2(B/2) : sec C sec2(C/2)= (1- sec A)csc2A : (1 - sec B)csc2B : (1 - sec C)csc2C
Barycentrics tan A sec2(A/2) : tan B sec2(B/2) : tan C sec2(C/2)
X(273) lies on these lines: 2,92 4,7 19,653 27,57 28,272 29,34 53,1086 75,225 78,322 108,675 226,469 269,1111 317,320 458,894
X(273) = isogonal conjugate of X(212)
X(273) = isotomic conjugate of X(78)
X(273) = X(I)-Ceva conjugate of X(J) for these (I,J): (264,342), (286,7), (331,92)
X(273) = cevapoint of X(I) and X(J) for these (I,J): (4,278), (34,57)
X(273) = X(I)-cross conjugate of X(J) for these (I,J): (4,92), (57,85), (225,278)
X(274) = ISOGONAL CONJUGATE OF X(213)
Trilinears b2c2/(b + c) : c2a2/(c + a) : a2b2/(a + b)= [a csc(A - ω)]/(b + c) : [b csc(B - ω)]/(c + a) :[c csc(C - ω)]/(a + b)
Barycentrics bc/(b + c) : ca/(c + a) : ab/(a + b)
X(274) lies on these lines:
1,75 2,39 7,959 10,291 21,99 28,242 57,85 58,870 69,443 81,239 88,799 110,767 183,474 213,894 264,475 278,331 315,377 325,442 961,1014
X(274) = isogonal conjugate of X(213)
X(274) = isotomic conjugate of X(37)
X(274) = complement of X(1655)
X(274) = X(310)-Ceva conjugate of X(314)
X(274) = cevapoint of X(I) and X(J) for these (I,J): (2,75), (85,348), (86,333)
X(274) = X(I)-cross conjugate of X(J) for these (I,J): (2,86), (75,310), (81,286), (333,314)
X(274) = crossdifference of evey pair of points on line X(669)X(798)
X(275) = CEVAPOINT OF ORTHOCENTER AND SYMMEDIAN POINT
Trilinears csc 2A sec(B - C) : csc 2B sec(C - A) : csc 2C sec(A - B)Barycentrics sec A sec(B - C) : sec B sec(C - A) : sec C sec(A - B)
X(275) lies on these lines:
2,95 4,54 13,472 14,473 17,471 18,470 25,262 51,107 53,288 76,276 83,297 94,324 98,427
X(275) = isogonal conjugate of X(216)
X(275) = isotomic conjugate of X(343)
X(275) = X(276)-Ceva conjugate of X(95)
X(275) = cevapoint of X(4) and X(6)
X(275) = X(I)-cross conjugate of X(J) for these (I,J): (6,54), (54,95)
X(275) = crosssum of X(217) and X(418)
X(276) = ISOGONAL CONJUGATE OF X(217)
Trilinears csc3A sec A sec(B - C) : csc3B sec B sec(C - A) : csc3C sec C sec(A - B)Barycentrics csc2A sec A sec(B - C) : csc2B sec B sec(C - A) : csc2C sec C sec(A - B)
X(276) lies on these lines: 3,95 4,327 54,290 76,275 97,401
X(276) = isogonal conjugate of X(217)
X(276) = isotomic conjugate of X(216)
X(276) = cevapoint of X(I) and X(J) for these (I,J): (2,264), (95,275)
X(276) = X(I)-cross conjugate of X(J) for these (I,J): (2,95), (401,290)
X(277) = ISOGONAL CONJUGATE OF X(218)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = [sec2(A/2)]/[- cos4A/2 + cos4B/2 + cos4C/2]
Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
X(277) lies on these lines: 1,142 3,105 7,218 57,169 220,1086 241,278 942,1002
X(277) = isogonal conjugate of X(218)
X(277) = isotomic conjugate of X(344)
X(277) = X(55)-cross conjugate of X(7)
X(278) = ISOGONAL CONJUGATE OF X(219)
Trilinears sec A tan A/2 : sec B tan B/2 : sec C tan C/2= csc A - 2 csc 2A : csc B - 2 csc 2B : csc C - 2 csc 2C
= (1 - sec A)/a : (1 - sec B)/b : (1 - sec C)/c
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/[(b + c - a)(b2 + c2 - a2)]
Barycentrics tan A tan A/2 : tan B tan B/2 : tan C tan C/2
= 1 - sec A : 1 - sec B : 1 - sec C
X(278) lies on these lines:
1,4 2,92 7,27 19,57 25,105 28,56 65,387 88,653 109,917 219,329 240,982 241,277 242,459 274,331 354,955 393,1108 412,962 443,1038 614,1096
X(278) = isogonal conjugate of X(219)
X(278) = isotomic conjugate of X(345)
X(278) = X(I)-Ceva conjugate of X(J) for these (I,J): (27,57), (92,196), (273,4), (331,7)
X(278) = cevapoint of X(19) and X(34)
X(278) = X(I)-cross conjugate of X(J) for these (I,J): (19,4), (56,7), (225,273)
X(279) = ISOGONAL CONJUGATE OF X(220)
Trilinears csc A tan2A/2 : csc B tan2B/2 : csc C tan2C/2= bc[a2 - (b - c)2]2 : ca[b2 - (c - a)2]2 : ab[c2 - (a - b)2]2
Barycentrics tan2A/2 : tan2B/2 : tan2C/2
X(279) lies on these lines: 1,7 2,85 28,1014 56,105 57,479 65,1002 144,220 145,664 304,346 942,955 985,1106
X(279) = isogonal conjugate of X(220)
X(279) = isotomic conjugate of X(346)
X(279) = cevapoint of X(57) and X(269)
X(279) = X(I)-cross conjugate of X(J) for these (I,J): (57,7), (269,479)
X(279) = crosssum of X(1) and X(170)
X(280) = X(1)-CROSS CONJUGATE OF X(8)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc2A/2)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(280) lies on these lines: 2,318 7,253 8,20 21,268 75,309 78,282 285,1043 341,345
X(280) = isogonal conjugate of X(221)
X(280) = isotomic conjugate of X(347)
X(280) = X(309)-Ceva conjugate of X(189)
X(280) = cevapoint of X(1) and X(84)
X(280) = X(I)-cross conjugate of X(J) for these (I,J): (1,8), (281,2), (282,189)
X(281) = X(37)-CROSS CONJUGATE OF X(9)
Trilinears sec A cot A/2 : sec B cot B/2 : sec C cot C/2= csc A + 2 csc 2A : csc B + 2 csc 2B : csc C + 2 csc 2C
= (1 + sec A)/a : (1 + sec B)/b : (1 + sec C)/c
Barycentrics tan A cot A/2 : tan B cot B/2 : tan C cot C/2
= 1 + sec A : 1 + sec B : 1 + sec C
X(281) lies on these lines:
1,282 2,92 4,9 7,653 8,29 28,958 33,200 37,158 45,53 48,944 100,1013 189,222 196,226 220,594 240,984 264,344 268,1012 318,346 380,950 451,1068 515,610 612,1096
X(281) = isogonal conjugate of X(222)
X(281) = isotomic conjugate of X(348)
X(281) = complement of X(347)
X(281) = X(I)-Ceva conjugate of X(J) for these (I,J): (29,33), (92,4)
X(281) = X(I)-cross conjugate of X(J) for these (I,J): (33,4), (37,9), (55,8)
X(281) = crosspoint of X(I) and X(J) for these (I,J): (2,280), (92,318)
X(281) = crosssum of X(I) and X(J) for these (I,J): (6,221), (48,603), (73,1409), (652,1364)
X(282) = X(6)-CROSS CONJUGATE OF X(9)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cot A/2)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(282) lies on the Thomson cubic and these lines:
1,281 2,77 3,9 4,3351 6,3341 19,102 48,947 57,3343 78,280 200,219 271,283 380,1036
X(282) = isogonal conjugate of X(223)
X(282) = X(189)-Ceva conjugate of X(84)
X(282) = X(I)-cross conjugate of X(J) for these (I,J): (6,9), (33,1)
X(282) = crosspoint of X(189) and X(280)
X(282) = crosssum of X(I) and X(J) for these (I,J): (6,1035), (198,221)
X(283) = X(3)-CROSS CONJUGATE OF X(21)
Trilinears (cos A)/(cos B + cos C) : (cos B)/(cos C + cos A) : (cos C)/(cos A + cos B)Barycentrics (sin 2A)/(cos B + cos C) : (sin 2B)/(cos C + cos A) : (sin 2C)/(cos A + cos B)
X(283) lies on these lines: 1,21 2,580 3,49 29,270 60,284 77,603 78,212 86,307 102,110 271,282 474,582 643,1043 859,945 1010,1065
X(283) = isogonal conjugate of X(225)
X(283) = X(333)-Ceva conjugate of X(284)
X(283) = cevapoint of X(I) and X(J) for these (I,J): (3,255), (212,219)
X(283) = X(3)-cross conjugate of X(21)
X(283) = crosspoint of X(332) and X(333)
X(284) = X(55)-CROSS CONJUGATE OF X(21)
Trilinears (sin A)/(cos B + cos C) : (sin B)/(cos C + cos A) : (sin C)/(cos A + cos B)Barycentrics a2/(cos B + cos C) : b2/(cos C + cos A) : c2/(cos A + cos B)
X(284) lies on these lines:
1,19 2,272 3,6 9,21 27,226 29,950 35,71 37,101 55,219 57,77 60,283 73,951 86,142 102,112 109,296 163,909 198,859 261,332 405,965 515,1065 942,1100
X(284) = isogonal conjugate of X(226)
X(284) = isotomic conjugate of X(349)
X(284) = inverse-in-Brocard-circle of X(579)
X(284) = X(I)-Ceva conjugate of X(J) for these (I,J): (81,58), (333,283)
X(284) = cevapoint of X(I) and X(J) for these (I,J): (6,48), (41,55)
X(284) = X(55)-cross conjugate of X(21)
X(284) = crosspoint of X(I) and X(J) for these (I,J): (21,81), (29,333)
X(284) = crosssum of X(I) and X(J) for these (I,J): (37,65), (73,1400)
X(284) = crossdifference of evey pair of points on line X(523)X(656)
X(285) = X(58)-CROSS CONJUGATE OF X(21)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = 1/[(cos B + cos C)(-1 - cos A + cos B + cos C)]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(285) lies on these lines: 21,84 29,81 271,282 280,1043
X(285) = isogonal conjugate of X(227)
X(285) = X(58)-cross conjugate of X(21)
X(286) = X(4)-CROSS CONJUGATE OF X(27)
Trilinears (csc 2A)/(sin B + sin C) : (csc 2B)/(sin C + sin A) : (csc 2C)/(sin A + sin B)Barycentrics (sec A)/(sin B + sin C) : (sec B)/(sin C + sin A) : (sec C)/(sin A + sin B)
X(286) lies on these lines: 4,69 7,331 19,27 28,242 29,34 99,915 112,767 158,969 322,1043
X(286) = isogonal conjugate of X(228)
X(286) = isotomic conjugate of X(72)
X(286) = cevapoint of X(I) and X(J) for these (I,J): (4,92), (7,273), (27,29), (28,81)
X(286) = X(I)-cross conjugate of X(J) for these (I,J): (4,27), (7,86), (81,274)
X(287) = X(2)-HIRST INVERSE OF X(98)
Trilinears cot A sec(A + ω) : cot B sec(B + ω) : cot C sec(C + ω)Barycentrics cos A sec(A + ω) : cos B sec(B + ω) : cos C sec(C + ω)
X(287) lies on these lines:
2,98 6,264 69,248 83,217 95,141 185,384 193,253 293,306 297,685 305,394 401,511 651,894 879,895
X(287) = reflection of X(648) in X(6)
X(287) = isogonal conjugate of X(232)
X(287) = isotomic conjugate of X(297)
X(287) = X(290)-Ceva conjugate of X(98)
X(287) = cevapoint of X(2) and X(401)
X(287) = X(248)-cross conjugate of X(98)
X(287) = X(2)-Hirst inverse of X(98)
X(288) = CEVAPOINT OF X(6) AND X(54)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [sec(B - C)]/[b cos(C - A) + c cos(B - A)]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(288) lies on these lines: 51,54 53,275 97,216
X(288) = isogonal conjugate of X(233)
X(288) = cevapoint of X(6) and X(54)
X(289) = ISOGONAL CONJUGATE OF X(236)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin A/2)/(cos B/2 + cos C/2 - cos A/2)Trilinears [1 - sin(A/2)] tan(A/2) : [1 - sin(B/2)] tan(B/2) : [1 - sin(C/2)] tan(C/2) (M. Iliev, 4/12/07)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(289) lies on these lines: 1,363 56,266 258,259
X(289) = isogonal conjugate of X(236)
X(289) = X(6)-cross conjugate of X(266)
X(289) = crosssum of X(1) and X(363)
X(290) = ISOGONAL CONJUGATE OF X(237)
Trilinears csc2A sec(A + ω) : csc2B sec(B + ω) : csc2C sec(C + ω)Barycentrics csc A sec(A + ω) : csc B sec(B + ω) : csc C sec(C + ω)
If you have The Geometer's Sketchpad, you can view the following dynamic sketch:
X(290).
X(290) lies on these lines:
2,327 3,76 6,264 54,276 66,317 67,340 68,315 69,670 71,190 72,668 73,336 248,385 265,316 308,311 892,895
X(290) = isogonal conjugate of X(237)
X(290) = isotomic conjugate of X(511)
X(290) = cevapoint of X(I) and X(J) for these (I,J): (2,511), (98,287)
X(290) = X(I)-cross conjugate of X(J) for these (I,J): (385,308), (401,276), (511,2)
X(291) = 2nd SHARYGIN POINT
Trilinears 1/(a2 - bc) : 1/(b2 - ca) : 1/(c2 - ab)Barycentrics a/(a2 - bc) : b/(b2 - ca) : c/(c2 - ab)
See the description at X(1281). The lines AD', BE', CF' defined there concur in X(256).
X(291) lies on these lines: 1,39 2,38 6,985 8,330 10,274 42,81 43,57 88,660 105,238 256,894 337,986 350,726 659,897 876,891
X(291) = reflection of X(I) in X(J) for these (I,J): (1,1015), (668,10)
X(291) = isogonal conjugate of X(238)
X(291) = isotomic conjugate of X(350)
X(291) = X(I)-cross conjugate of X(J) for these (I,J): (239,256), (518,1)
X(291) = X(I)-Hirst inverse of X(J) for these (I,J): (1,292), (2,335)
X(292) = X(1)-HIRST INVERSE OF X(291)
Trilinears a/(a2 - bc) : b/(b2 - ca) : c/(c2 - ab)Barycentrics a2/(a2 - bc) : b2/(b2 - ca) : c2/(c2 - ab)
X(292) lies on these lines: 1,39 2,334 6,869 9,87 37,86 44,660 58,101 106,813 171,893 269,1020 659,665
X(292) = isogonal conjugate of X(239)
X(292) = isotomic conjugate of X(1921)
X(292) = X(335)-Ceva conjugate of X(295)
X(292) = cevapoint of X(171) and X(238)
X(292) = crossdifference of evey pair of points on line X(659)X(812)
X(292) = X(1)-Hirst inverse of X(291)
X(293) = ISOGONAL CONJUGATE OF X(240)
Trilinears cos A sec(A + ω) : cos B sec(B + ω) : cos C sec(C + ω)Barycentrics sin 2A sec(A + ω) : sin 2B sec(B + ω) : sin 2C sec(C + ω)
X(293) lies on these lines: 1,163 31,92 72,248 98,109 255,304 287,306
X(293) = isogonal conjugate of X(240)
X(294) = ISOGONAL CONJUGATE OF X(241)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b2 + c2 - ab - ac)Barycentrics af(a,b,c) : bf(b,c,a) :cf(c,a,b)
X(294) lies on these lines: 1,41 2,949 4,218 6,7 8,220 19,1041 84,580 104,919 239,666 241,910 314,645
X(294) = isogonal conjugate of X(241)
X(294) = crosssum of X(I) and X(J) for these (I,J): (672,1458), (910,1279)
X(294) = crossdifference of evey pair of points on line X(926)X(1362)
X(294) = X(1)-Hirst inverse of X(105)
X(295) = ISOGONAL CONJUGATE OF X(242)
Trilinears (cos A)/(a2 - bc) : (cos B)/(b2 - ca) : (cos C)/(c2 - ab)Barycentrics (sin 2A)/(a2 - bc) : (sin 2B)/(b2 - ca) : (sin 2C)/(c2 - ab)
X(295) lies on these lines: 27,335 43,57 58,101 72,337 103,813 150,334 875,926 876,928
X(295) = isogonal conjugate of X(242)
X(295) = X(335)-Ceva conjugate of X(292)
X(295) = crosspoint of X(335) and X(337)
X(296) = ISOGONAL CONJUGATE OF X(243)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (cos A)/[cos2A - cos B cos C]
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = (sin 2A)/[cos2A - cos B cos C]
X(296) lies on these lines: 1,185 3,820 29,65 109,284
X(296) = isogonal conjugate of X(243)
X(297) = X(2)-HIRST INVERSE OF X(4)
Trilinears csc 2A cos(A + ω) : csc 2B cos(B + ω) : csc 2C cos(C + ω)Barycentrics sec A cos(A + ω) : sec B cos(B + ω) : sec C cos(C + ω)
X(297) lies on these lines:
2,3 6,317 53,141 69,393 76,343 83,275 92,257 232,325 249,316 287,685 315,394 340,524 525,850
X(297) = midpoint of X(340) and X(648)
X(297) = reflection of X(401) in X(441)
X(297) = isogonal conjugate of X(248)
X(297) = isotomic conjugate of X(287)
X(297) = inverse-in-orthocentroidal-circle of X(458)
X(297) = complement of X(401)
X(297) = anticomplement of X(441)
X(297) = cevapoint of X(232) and X(511)
X(297) = X(511)-cross conjugate of X(325)
X(297) = crossdifference of evey pair of points on line X(184)X(647)
X(297) = X(I)-Hirst inverse of X(J) for (I,J) = (2,4), (193,1249)
Centers 298- 350
are isotomic conjugates of previously listed centers.
X(298) ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER
Trilinears csc2A sin(A + π/3) : csc2B sin(B + π/3) : csc2C sin(C + π/3)Barycentrics csc A sin(A + π/3) : csc B sin(B + π/3) : csc C sin(C + π/3)
X(298) lies on these lines:
2,6 3,617 5,634 13,532 14,76 15,533 18,636 99,531 140,628 264,472 316,530 317,473 319,1082 340,470 381,622 511,1080
X(298) = midpoint of X(616) and X(621)
X(298) = reflection of X(I) in X(J) for these (I,J): (13,623), (15,618), (299,325), (385,395)
X(298) = isogonal conjugate of X(3457)
X(298) = isotomic conjugate of X(13)
X(298) = complement of X(3180)
X(298) = anticomplement of X(396)
X(298) = X(300)-Ceva conjugate of X(303)
X(298) = X(15)-cross conjugate of X(470)
X(298) = X(2)-Hirst inverse of X(299)
X(299) = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER
Trilinears csc2A sin(A - π/3) : csc2B sin(B - π/3) : csc2C sin(C - π/3)Barycentrics csc A sin(A - π/3) : csc B sin(B - π/3) : csc C sin(C - π/3)
X(299) lies on these lines:
2,6 3,616 5,633 13,76 14,533 16,532 17,635 30,617 75,554 99,530 140,627 264,473 316,531 317,472 319,559 340,471 381,621 383,511
X(299) = midpoint of X(617) and X(622)
X(299) = reflection of X(I) in X(J) for these (I,J): (14,624), (16,619), (298,325), (385,396)
X(2998) = isogonal conjugate of X(3458)
X(299) = isotomic conjugate of X(14)
X(299) = complement of X(3181)
X(299) = anticomplement of X(395)
X(299) = X(301)-Ceva conjugate of X(302)
X(299) = X(16)-cross conjugate of X(471)
X(299) = X(2)-Hirst inverse of X(298)
X(300) ISOTOMIC CONJUGATE OF 1st ISODYNAMIC CENTER
Trilinears csc2A csc(A + π/3) : csc2B csc(B + π/3) : csc2C csc(C + π/3)Barycentrics csc A csc(A + π/3) : csc B csc(B + π/3) : csc C csc(C + π/3)
X(300) lies on these lines: 2,94 13,76 264,302 265,621 303,311
X(300) = isotomic conjugate of X(15)
X(300) = cevapoint of X(298) and X(303)
X(300) = X(94)-Hirst inverse of X(301)
X(301) ISOTOMIC CONJUGATE OF 2nd ISODYNAMIC CENTER
Trilinears csc2A csc(A - π/3) : csc2B csc(B - π/3) : csc2C csc(C - π/3)Barycentrics csc A csc(A - π/3) : csc B csc(B - π/3) : csc C csc(C - π/3)
X(301) lies on these lines: 2,94 14,76 264,303 265,622 302,311
X(301) = isotomic conjugate of X(16)
X(301) = cevapoint of X(299) and X(302)
X(301) = X(94)-Hirst inverse of X(300)
X(302) ISOTOMIC CONJUGATE OF 1st NAPOLEON POINT
Trilinears csc2A sin(A + π/6) : csc2B sin(B + π/6) : csc2C sin(C + π/6)Barycentrics csc A sin(A + π/6) : csc B sin(B + π/6) : csc C sin(C + π/6)
If you have The Geometer's Sketchpad, you can view X(302).
X(302) lies on these lines:
2,6 3,621 5,622 14,99 16,316 18,76 61,629 140,633 264,300 301,311 317,470 381,616 549,617
X(302) = isotomic conjugate of X(17)
X(302) = X(301)-Ceva conjugate of X(299)
X(302) = X(61)-cross conjugate of X(473)
X(303) ISOTOMIC CONJUGATE OF 2nd NAPOLEON POINT
Trilinears csc2A sin(A - π/6) : csc2B sin(B - π/6) : csc2C sin(C - π/6)Barycentrics csc A sin(A - π/6) : csc B sin(B - π/6) : csc C sin(C - π/6)
If you have The Geometer's Sketchpad, you can view X(303).
X(303) lies on these lines:
2,6 3,622 5,621 13,99 15,316 17,76 62,630 140,634 264,301 300,311 317,471 381,617 549,616
X(303) = isotomic conjugate of X(18)
X(303) = X(300)-Ceva conjugate of X(298)
X(303) = X(62)-cross conjugate of X(472)
X(304) = ISOTOMIC CONJUGATE OF X(19)
Trilinears (cot A)csc2A : (cot B)csc2B : (cot C)csc2C= cos A csc(A - ω) : cos B csc(B - ω) : cos C csc(C - ω)
Barycentrics (cos A)csc2A : (cos B)csc2B : (cos C)csc2C
X(304) lies on these lines:
1,75 63,1102 69,72 76,85 92,561 255,293 279,346 305,306 309,322 341,1088 345,348
X(304) = isogonal conjugate of X(1973)
X(304) = isotomic conjugate of X(19)
X(304) = cevapoint of X(I) and X(J) for these (I,J): (63,326), (69,345), (312,322)
X(304) = X(I)-cross conjugate of X(J) for these (I,J): (63,75), (306,69)
X(305) = ISOTOMIC CONJUGATE OF X(25)
Trilinears b4c4cos A : c4a4cos B : a4b4cos C= cot A csc(A - ω) : cot B csc(B - ω) : cot C csc(C - ω)
Barycentrics b3c3cos A : c3a3cos B : a3b3cos C
X(305) lies on these lines:
2,39 22,99 25,683 95,183 264,325 287,394 304,306 311,1007 341,1088 350,614
X(305) = isogonal conjugate of X(1974)
X(305) = isotomic conjugate of X(25)
X(305) = anticomplement of X(1196)
X(305) = X(69)-cross conjugate of X(76)
X(306) = ISOTOMIC CONJUGATE OF X(27)
Trilinears (b2c2)(b + c)cos A : (c2a2)(c + a)cos B : (a2b2)(a + b)cos CBarycentrics bc(b + c)cos A : ca(c + a)cos B : ab(a + b)cos C
X(306) lies on these lines:
1,2 27,1043 63,69 72,440 92,264 209,518 226,321 253,329 287,293 304,305 319,333
X(306) = isogonal conjugate of X(1474)
X(306) = isotomic conjugate of X(27)
X(306) = complement of X(3187)
X(306) = X(I)-Ceva conjugate of X(J) for these (I,J): (69, 72), (312,321), (313,10)
X(306) = X(I)-cross conjugate of X(J) for these (I,J): (71,10), (72,307), (440,2)
X(306) = crosspoint of X(I) and X(J) for these (I,J): (69,304), (312,345)
X(306) = crosssum of X(604) and X(608)
X(307) = ISOTOMIC CONJUGATE OF X(29)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2c2(b + c)(cos A)/(b + c - a)Barycentrics af(a,b,c) : bf(b,c,a) :cf(c,a,b)
X(307) lies on these lines: 2,7 8,253 69,73 75,225 86,283 95,320 141,241 269,936 319,664 948,966
X(307) = isogonal conjugate of X(2299)
X(307) = isotomic conjugate of X(29)
X(307) = X(349)-Ceva conjugate of X(226)
X(307) = X(I)-cross conjugate of X(J) for these (I,J): (72,306), (73,226)
X(307) = crosspoint of X(69) and X(75)
X(307) = crosssum of X(25) and X(31)
X(308) = ISOTOMIC CONJUGATE OF X(39)
Trilinears b3c3/(b2 + c2) : c3a3/(c2 + a2) : a3b3/(a2 + b2)= csc2A csc(A + ω) : csc2B csc(B + ω) : csc2C csc(C + ω)
= [csc(A - ω)]/(b2 + c2) : [csc(B - ω)]/(c2 + a2) : [csc(C - ω)]/(a2 + b2)
Barycentrics (b2c2)/(b2 + c2) : (c2a2)/(c2 + a2) : (a2b2)/(a2 + b2)
= csc A csc(A + ω) : csc B csc(B + ω) : csc C csc(C + ω)
X(308) lies on these lines: 2,702 6,76 25,183 42,313 69,263 111,689 141,670 251,385 290,311
X(308) = isogonal conjugate of X(3051)
X(308) = isotomic conjugate of X(39)
X(308) = cevapoint of X(2) and X(76)
X(308) = X(I)-cross conjugate of X(J) for these (I,J): (2,83), (385,290)
X(309) = ISOTOMIC CONJUGATE OF X(40)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc2A)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(309) lies on these lines: 69,189 75,280 77,318 84,314 85,264 304,322
X(309) = isogonal conjugate of X(2187)
X(309) = isotomic conjugate of X(40)
X(309) = cevapoint of X(189) and X(280)
X(309) = X(I)-cross conjugate of X(J) for these (I,J): (7,75), (92,85)
X(310) = ISOTOMIC CONJUGATE OF X(42)
Trilinears b3c3/(b + c) : c3a3/(c + a) : a3b3/(a + b)Barycentrics b2c2/(b + c) : c2a2/(c + a) : a2b2/(a + b)
X(310) lies on these lines: 2,39 7,314 38,75 86,350 99,675 261,272 321,335 333,673 670,903 871,982
X(310) = isogonal conjugate of X(1918)
X(310) = isotomic conjugate of X(42)
X(310) = cevapoint of X(I) and X(J) for these (I,J): (75,76), (274,314)
X(310) = X(75)-cross conjugate of X(274)
X(311) = ISOTOMIC CONJUGATE OF X(54)
Trilinears csc2A cos(B - C) : csc2B cos(C - A) : csc2C cos(A - B)Barycentrics csc A cos(B - C) : csc B cos(C - A) : csc C cos(A - B)
X(311) lies on these lines: 2,570 4,69 22,157 53,324 95,99 141,338 290,308 300,303 301,302 305,1007
X(311) = isotomic conjugate of X(54)
X(311) = anticomplement of X(570)
X(311) = X(76)-Ceva conjugate of X(343)
X(311) = cevapoint of X(5) and X(343)
X(311) = X(5)-cross conjugate of X(324)
X(312) = ISOTOMIC CONJUGATE OF X(57)
Trilinears (b + c - a)b2c2 : (c + a - b)c2a2 : (a + b - c)a2b2= (1 + cos A)csc(A - ω) : (1 + cos B)csc(B - ω) : (1 + cos C)csc(C - ω)
Trilinears (csc A)/(1 - cos A) : (csc B)/(1 - cos B) : (csc C)/(1 - cos C) (M. Iliev, 4/12/07)
Barycentrics bc(b + c - a) : ca(c + a - b) : ab(a + b - c)
X(312) lies on these lines: 1,1089 2,37 8,210 9,314 29,33 63,190 69,189 76,85 92,264 212,643 223,664 726,982 894,940 975,1010
X(312) = isogonal conjugate of X(604)
X(312) = isotomic conjugate of X(57)
X(312) = complement of X(3210)
X(312) = X(I)-Ceva conjugate of X(J) for these (I,J): (76,75), (304,322), (314,8)
X(312) = cevapoint of X(I) and X(J) for these (I,J): (2,329), (8,346), (9,78), (306,321)
X(312) = X(I)-cross conjugate of X(J) for these (I,J): (8,75), (9,318), (306,345), (346,341)
X(312) = crosssum of X(I) and X(J) for these (I,J): (32,1397), (56,1403), (57,1424)
X(313) = ISOTOMIC CONJUGATE OF X(58)
Trilinears (b + c)b3c3 : (c + a)c3a3 : (a + b)a3b3= (b + c)csc(A - ω) : (c + a)csc(B - ω) : (a + b)csc(C - ω)
Barycentrics (b + c)b2c2 : (c + a)c2a2 : (a + b)a2b2
X(313) lies on these lines: 10,75 12,349 42,308 71,190 80,314 92,264 321,594 561,696
X(313) = isogonal conjugate of X(2206)
X(313) = isotomic conjugate of X(58)
X(313) = X(76)-Ceva conjugate of X(321)
X(313) = cevapoint of X(10) and X(306)
X(313) = X(321)-cross conjugate of X(349)
X(313) = crosssum of X(32) and X(560)
X(314) = ISOTOMIC CONJUGATE OF X(65)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2c2(b + c - a)/(b + c)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc(b + c - a)/(b + c)
X(314) lies on these lines:
1,75 2,941 4,69 6,981 7,310 9,312 21,261 29,1039 58,987 79,320 80,313 81,321 84,309 99,104 256,350 294,645
X(314) = isogonal conjugate of X(1402)
X(314) = isotomic conjugate of X(65)
X(314) = anticomplement of X(2092)
X(314) = X(310)-Ceva conjugate of X(274)
X(314) = cevapoint of X(I) and X(J) for these (I,J): (8,312), (69,75)
X(314) = X(I)-cross conjugate of X(J) for these (I,J): (8,333), (69,332), (333,274), (497,29)
X(315) = ISOTOMIC CONJUGATE OF X(66)
Trilinears bc(b4 + c4 - a4) : ca(c4 + a4 - b4) : ab(a4 + b4 - c4)Barycentrics b4 + c4 - a4 : c4 + a4 - b4 : a4 + b4 - c4
X(315) lies on these lines:
2,32 3,325 4,69 5,183 8,760 20,99 68,290 192,746 194,736 274,377 297,394 343,458 371,491 372,492 631,1007
X(315) = midpoint of X(637) and X(638)
X(315) = reflection of X(I) in X(J) for these (I,J): (32,626), (371,640, (372,639)
X(315) = isogonal conjugate of X(2353)
X(315) = isotomic conjugate of X(66)
X(315) = anticomplement of X(32)
X(315) = anticomplementary conjugate of X(194)
X(315) = X(I)-cross conjugate of X(J) for these (I,J): (206,2)
X(316) = DROUSSENT PIVOT
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b4 + c4 - a4 - b2c2)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b4 + c4 - a4 - b2c2
The reflection of X(99) in the polar of X(76).
Lucien Droussent, "Cubiques circulaires anallagmatiques par points réciproques ou isogonaux," Mathesis 62 (1953) 204-215.
X(316) lies on these lines:
2,187 4,69 15,303 16,302 30,99 115,385 148,538 183,381 249,297 265,290 298,530 299,531 376,1007 384,626 512,850 524,671 691,858
X(316) = midpoint of X(621) and X(622)
X(316) = reflection of X(I) in X(J) for these (I,J): (15,624), (16,623), (99,325), (385,115), (691,858)
X(316) = isogonal conjugate of X(3455)
X(316) = isotomic conjugate of X(67)
X(316) = anticomplement of X(187)
X(316) = crosssum of X(39) and X(187)
X(317) = ISOTOMIC CONJUGATE OF X(68)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A cos 2A csc2A= bc cot 2A : ca cot 2B : ab cot 2C
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = tan A cos 2A csc2A
X(317) lies on these lines:
2,95 4,69 6,297 25,325 53,524 66,290 141,458 183,427 193,393 261,406 273,320 298,473 299,472 302,470 303,471 318,319 459,1007
X(317) = isogonal conjugate of X(2351)
X(317) = isotomic conjugate of X(68)
X(317) = anticomplement of X(577)
X(317) = cevapoint of X(52) and X(467)
X(318) = ISOTOMIC CONJUGATE OF X(77)
Trilinears (1 + sec A)/a2 : (1 + sec B)/b2 : (1 + sec C)/c2= sec A csc2A/2 : sec B csc2B/2 : sec C csc2C/2
Barycentrics (1 + sec A)/a : (1 + sec B)/b : (1 + sec C)/c
X(318) lies on these lines:
2,280 4,8 10,158 29,33 53,594 63,412 75,225 77,309 108,404 200,1089 208,653 239,458 243,958 253,342 281,346 317,319 475,1068
X(318) = isogonal conjugate of X(603)
X(318) = isotomic conjugate of X(77)
X(318) = X(264)-Ceva conjugate of X(92)
X(318) = cevapoint of X(9) and X(33)
X(318) = X(I)-cross conjugate of X(J) for these (I,J): (9,312), (10,8), (281,92)
X(319) = ISOTOMIC CONJUGATE OF X(79)
Trilinears (1 + 2 cos A)/a2 : (1 + 2 cos B)/b2 : (1 + 2 cos C)/c2Barycentrics (1 + 2 cos A)/a : (1 + 2 cos B)/b : (1 + 2 cos C)/c
X(319) lies on these lines: 2,1100 7,8 10,86 80,313 141,239 171,757 200,326 261,502 298,1082 299,559 306,333 307,664 317,318 321,1029 344,391 524,594
X(319) = reflection of X(894) in X(594)
X(319) = isotomic conjugate of X(79)
X(319) = anticomplement of X(1100)
X(320) = ISOTOMIC CONJUGATE OF X(80)
Trilinears (1 - 2 cos A)/a2 : (1 - 2 cos B)/b2 : (1 - 2 cos C)/c2Barycentrics (1 - 2 cos A)/a : (1 - 2 cos B)/b : (1 - 2 cos C)/c
X(320) lies on these lines:
1,752 2,44 7,8 58,86 79,314 95,307 141,894 144,344 190,527 239,524 269,326 273,317 334,660 350,513 519,679
X(320) = reflection of X(239) in X(1086)
X(320) = isotomic conjugate of X(80)
X(320) = X(214)-cross conjugate of X(1)
X(320) = crosssum of X(42) and X(902)
X(321) = ISOTOMIC CONJUGATE OF X(81)
Trilinears (b + c)b2c2 : (c + a)c2a2 : (a + b)a2b2= a(b + c)csc(A - ω) : b(c + a)csc(B - ω) : c(a + b)csc(C - ω)
Barycentrics bc(b + c) : ca(c + a) : ab(a + b)
X(321) lies on these lines:
1,964 2,37 4,8 10,756 38,726 76,561 81,314 83,213 98,100 190,333 226,306 310,335 313,594 319,1029 668,671 693,824
X(321) = reflection of X(42) in X(1215)
X(321) = isogonal conjugate of X(1333)
X(321) = isotomic conjugate of X(81)
X(321) = X(I)-Ceva conjugate of X(J) for (I,J) = (75,10), (76,313), (312,306)
X(321) = cevapoint of X(37) and X(72)
X(321) = X(442)-cross conjugate of X(264)
X(321) = crosspoint of X(I) and X(J) for these (I,J): (75,76), (313,349)
X(321) = crosssum of X(31) and X(32)
X(321) = crossdifference of evey pair of points on line X(667)X(838)
X(322) = ISOTOMIC CONJUGATE OF X(84)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (-1 - cos A + cos B + cos C)csc2ABarycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (-1 - cos A + cos B + cos C)csc A
X(322) lies on these lines: 2,1108 7,8 78,273 92,264 227,347 253,341 286,1043 304,309 326,664
X(322) = isogonal conjugate of X(2208)
X(322) = isotomic conjugate of X(84)
X(322) = anticomplement of X(1108)
X(322) = X(304)-Ceva conjugate of X(312)
X(322) = X(347)-cross conjugate of X(75)
X(323) = ISOTOMIC CONJUGATE OF X(94)
Trilinears sin 3A csc2A : sin 3B csc2B : sin 3C csc2CTrilinears g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 4 sin A - 3 csc A
Barycentrics sin 3A csc A : sin 3B csc B : sin 3C csc C
X(323) lies on these lines: 2,6 20,155 23,110 30,146 140,195 187,249 401,525
X(323) = reflection of X(23) in X(110)
X(323) = isogonal conjugate of X(1989)
X(323) = isotomic conjugate of X(94)
X(323) = X(340)-Ceva conjugate of X(186)
X(323) = cevapoint of X(6) and X(399)
X(323) = X(50)-cross conjugate of X(186)
X(323) = crosssum of X(395) and X(396)
X(323) = crossdifference of evey pair of points on line X(51)X(512)
X(324) = ISOTOMIC CONJUGATE OF X(97)
Trilinears bc sec A cos(B - C) : ca sec B cos(C - A) : ab sec C cos(A - B)Barycentrics sec A cos(B - C) : sec B cos(C - A) : sec C cos(A - B)
X(324) lies on these lines: 2,216 4,52 53,311 94,275 110,436 143,565
X(324) = isotomic conjugate of X(97)
X(324) = X(264)-Ceva conjugate of X(5)
X(324) = cevapoint of X(I) and X(J) for these (I,J): (5,53), (52,216)
X(324) = X(5)-cross conjugate of X(311)
X(325) = X(2)-HIRST INVERSE OF X(69)
Trilinears csc2A cos(A + ω) : csc2B cos(B + ω) : csc2C cos(C + ω)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b4 + c4 - a2b2 - a2c2)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b4 + c4 - a2b2 - a2c2
X(325) lies on these lines:
2,6 3,315 5,76 11,350 22,160 25,317 30,99 39,626 114,511 115,538 187,620 232,297 250,340 264,305 274,442 383,622 523,684 621,1080
X(325) = midpoint of X(I) and X(J) for these (I,J): (99,316), (298,299)
X(325) = reflection of X(I) in X(J) for these (I,J): (115,625), (187,620), (385,230), (1513,114)
X(325) = isogonal conjugate of X(1976)
X(325)= isotomic conjugate of X(325)
X(325) = complement of X(98)
X(325) = anticomplement of X(230)
X(325) = cevapoint of X(2) and X(147)
X(325) = X(I)-cross conjugate of X(J) for these (I,J): (114,2), (511,297)
X(325) = crossdifference of evey pair of points on line X(32)X(512)
X(325) = X(2)-Hirst inverse of X(69)
X(326) = ISOTOMIC CONJUGATE OF X(158)
Trilinears cot2A : cot2B : cot2CBarycentrics csc A - sin A : csc B - sin B : csc C - sin C
X(326) lies on these lines: 1,75 48,63 69,73 200,319 255,1102 269,320 322,664 610,662
X(326) = isogonal conjugate of X(1096)
X(326) = isotomic conjugate of X(158)
X(326) = X(I)-Ceva conjugate of X(J) for these (I,J): (304,63), (332,69)
X(326) = X(255)-cross conjugate of X(63)
X(327) = ISOTOMIC CONJUGATE OF X(182)
Trilinears csc2A sec(A - ω) : csc2B sec(B - ω) : csc2C sec(C - ω)= sin A csc(2A - 2 ω): sin B csc(2B - 2 ω) : sin C csc(2C - 2 ω)
Barycentrics csc A sec(A - ω) : csc B sec(B - ω) : csc C sec(C - ω)
X(327) lies on these lines: 2,290 4,276 5,76 53,141 69,263 95,160
X(327) = isotomic conjugate of X(182)
X(328) = ISOTOMIC CONJUGATE OF X(186)
Trilinears cot A csc 3A : cot B csc 3B : cot C csc 3CBarycentrics cos A csc 3A : cos B csc 3B : cos C csc 3C
X(328) lies on these lines: 2,94 69,265 95,99
X(328) = isotomic conjugate of X(186)
X(328) = X(265)-cross conjugate of X(94)
X(329) = ISOTOMIC CONJUGATE OF X(189)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = ( -1 - cos A + cos B + cos C)(csc A)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = -1 - cos A + cos B + cos C
X(329) lies on these lines:
1,452 2,7 4,8 20,78 55,1005 69,189 100,972 190,345 191,498 196,342 200,516 219,278 220,948 223,347 253,306 388,960 392,1056 394,651 405,999 497,518
X(329) = isogonal conjugate of X(1436)
X(329) = isotomic conjugate of X(189)
X(329) = cyclocevian conjugate of X(1034)
X(329) = anticomplement of X(57)
X(329) = anticomplementary conjugate of X(7)
X(329) = X(I)-Ceva conjugate of X(J) for (I,J) = (69,8), (312,2)
X(329) = X(I)-cross conjugate of X(J) for these (I,J): (40,347), (223,2)
X(330) = ISOTOMIC CONJUGATE OF X(192)
Trilinears bc/(ab + ac - bc) : ca/(bc + ba - ca) : ab/(ca + cb - ab)Barycentrics 1/(ab + ac - bc) : 1/(bc + ba - ca) : 1/(ca + cb - ab)
X(330) lies on these lines: 1,87 2,1107 8,291 56,385 57,239 76,1015 105,932 145,1002 193,959 257,982
X(330) = isogonal conjugate of X(2176)
X(330) = isotomic conjugate of X(192)
X(330) = X(87)-Ceva conjugate of X(2)
X(330) = X(75)-cross conjugate of X(2)
X(331) = ISOTOMIC CONJUGATE OF X(219)
Trilinears sec2(A/2) csc(2A) : sec2(B/2) csc(2B) : sec2(C/2) csc(2C)= (1 - sec A)csc(A - ω) : (1 - sec B)csc(B - ω) : (1 - sec C)csc(C - ω)
Barycentrics sec2(A/2) sec A : sec2(B/2) sec B : sec2(C/2) sec C
X(331) lies on these lines: 4,150 7,286 34,870 75,225 85,92 108,767 274,278
X(331) = isotomic conjugate of X(219)
X(331) = cevapoint of X(I) and X(J) for these (I,J): (7,278), (92,273)
X(331) = X(92)-cross conjugate of X(264)
X(332) = ISOTOMIC CONJUGATE OF X(215)
Trilinears (cot A csc A)/(cos B + cos C) : (cot B csc B)/(cos C + cos A) : (cot C csc C)/(cos A + cos B)Barycentrics (cot A)/(cos B + cos C) : (cot B)/(cos C + cos A) : (cot C)/(cos A + cos B)
X(332) lies on these lines: 1,75 3,69 21,1036 99,102 219,345 261,284 1014,1037
X(332) = isotomic conjugate of X(225)
X(332) = cevapoint of X(I) and X(J) for these (I,J): (69,326), (78,345)
X(332) = X(I)-cross conjugate of X(J) for these (I,J): (69,314), (283,333)
X(333) CEVAPOINT OF X(8) AND X(9)
Trilinears bc(b + c - a)/(b + c) : ca(c + a - b)/(c + a) : ab(a + b - c)/(a + b)Barycentrics (b + c - a)/(b + c) : (c + a - b)/(c + a) : (a + b - c)/(a + b)
X(333) lies on these lines:
2,6 8,21 9,312 10,58 19,27 29,270 57,85 190,321 239,257 261,284 306,319 310,673 662,909 740,846 859,956 1021,1024
X(333) = isogonal conjugate of X(1400)
X(333) = isotomic conjugate of X(226)
X(333) = X(I)-Ceva conjugate of X(J) for these (I,J): (261,21), (274,86)
X(333) = cevapoint of X(I) and X(J) for these (I,J): (2,63), (8,9), (283,284)
X(333) = X(I)-cross conjugate of X(J) for these (I,J): (8,314), (9,21), (21,86), (283,332), (284,29)
X(333) = crosspoint of X(274) and X(314)
X(333) = crosssum of X(213) and X(1402)
X(333) = crossdifference of evey pair of points on line X(512)X(810)
X(334) = ISOTOMIC CONJUGATE OF X(238)
Trilinears b2c2/(a2 - bc) : c2a2/(b2 - ca) : a2b2/(c2 - ab)Barycentrics bc/(a2 - bc) : ca/(b2 - ca) : ab/(c2 - ab)
X(334) lies on these lines: 2,292 10,274 12,85 75,141 76,1089 150,295 320,660 741,839 767,813
X(334) = isogonal conjugate of X(2210)
X(334) = isotomic conjugate of X(238)
X(334) = X(75)-Hirst inverse of X(335)
X(335) = ISOTOMIC CONJUGATE OF X(239)
Trilinears bc/(a2 - bc) : ca/(b2 - ca) : ab/(c2 - ab)Barycentrics 1/(a2 - bc) : 1/(b2 - ca) : 1/(c2 - ab)
X(335) lies on these lines: 1,384 2,38 7,192 27,295 37,86 75,141 76,871 239,518 257,694 310,321 320,742 536,903 675,813 741,835 876,900
X(335) = reflection of X(I) in X(J) for these (I,J): (75,1086), (190,37)
X(335) = isogonal conjugate of X(1914)
X(335) = isotomic conjugate of X(239)
X(335) = cevapoint of X(I) and X(J) for these (I,J): (37,518), (292,295)
X(335) = X(I)-cross conjugate of X(J) for these (I,J): (295,337), (350,257)
X(335) = X(I)-Hirst inverse of X(J) for these (I,J): (2,291), (75,334), (292,894)
X(336) = ISOTOMIC CONJUGATE OF X(240)
Trilinears csc A cot A sec(A + ω) : csc B cot B sec(B + ω) : csc C cot C sec(C + ω)Barycentrics cot A sec(A + ω) : cot B sec(B + ω) : cot C sec(C + ω)
X(336) lies on these lines: 1,811 48,75 73,290 255,293
X(336) = isotomic conjugate of X(240)
X(337) = ISOTOMIC CONJUGATE OF X(242)
Trilinears (csc A cot A)/(a2 - bc) : (csc B cot B)/(b2 - ca) : (csc C cot C)/(c2 - ab)Barycentrics (cot A)/(a2 - bc) : (cot B)/(b2 - ca) : (cot C)/(c2 - ab)
X(337) lies on these lines: 12,85 37,86 72,295 201,348 291,986
X(337) = isotomic conjugate of X(242)
X(337) = X(295)-cross conjugate of X(335)
X(338) CEVAPOINT OF X(115) AND X(125)
Trilinears (b2 - c2)2/a3 : (c2 - a2)2/b3 : (a2 - b2)2/c3= csc A sin2(B - C) : csc B sin2(C - A) : csc C sin2(A - B)
Barycentrics (b2 - c2)2/a2 : (c2 - a2)2/b2 : (a2 - b2)2/c2
= sin2(B - C) : sin2(C - A) : sin2(A - B)
X(338) lies on these lines:
2,94 4,67 6,264 50,401 76,599 115,127 125,136 141,311
X(338) = isotomic conjugate of X(249)
X(338) = X(264)-Ceva conjugate of X(523)
X(338) = cevapoint of X(115) and X(125)
X(338) = X(125)-cross conjugate of X(339)
X(339) = ISOTOMIC CONJUGATE OF X(250)
Trilinears (b2 - c2)2(cos A)/a4 : (c2 - a2)2(cos B)/b4 : (a2 - b2)2(cos C)/c4= csc A cot A sin2(B - C) : csc B cot B sin2(C - A) : csc C cot C sin2(A - B)
Barycentrics (b2 - c2)2(cos A)/a3 : (c2 - a2)2(cos B)/b3 : (a2 - b2)2(cos C)/c3
= cot A sin2(B - C) : cot B sin2(C - A) : cot C sin2(A - B)
X(339) lies on these lines: 3,76 69,265 115,127 264,381
X(339) = isotomic conjugate of X(250)
X(339) = X(76)-Ceva conjugate of X(525)
X(339) = X(125)-cross conjugate of X(338)
X(340) = ISOTOMIC CONJUGATE OF X(265)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = sec A sin 3A csc3A
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = sec A sin 3A csc2A
X(340) lies on these lines: 4,69 67,290 95,140 250,325 297,524 298,470 299,471 447,540 458,599 520,850
X(340) = reflection of X(648) in X(297)
X(340) = isotomic conjugate of X(265)
X(340) = anticomplement of X(3284)
X(340) = cevapoint of X(186) and X(323)
X(341) = ISOTOMIC CONJUGATE OF X(269)
Trilinears b2c2(b + c - a)2 : c2a2(c + a - b)2 : a2b2(a + b - c)2= csc4A/2 : csc4B/2 : csc4C/2
Barycentrics bc(b + c - a)2 : ca(c + a - b)2 : ab(a + b - c)2
X(341) lies on these lines: 1,1050 8,210 10,75 40,190 200,1043 253,322 280,345 304,668 305,1088
X(341) = isogonal conjugate of X(1106)
X(341) = isotomic conjugate of X(269)
X(341) = X(346)-cross conjugate of X(312)
X(342) = ISOTOMIC CONJUGATE OF X(271)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc 2A tan A/2)(1 + cos A - cos B - cos C)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sec A tan A/2)(1 + cos A - cos B - cos C)
X(342) lies on these lines: 4,7 9,653 85,264 92,226 108,1005 196,329 253,318 393,948
X(342) = isogonal conjugate of X(2188)
X(342) = isotomic conjugate of X(271)
X(342) = X(I)-Ceva conjugate of X(J) for these (I,J): (85,92), (264,273)
X(342) = cevapoint of X(208) and X(223)
X(343) = ISOTOMIC CONJUGATE OF X(275)
Trilinears cot A cos(B - C) : cot B cos(C - A) : cot C cos(A - B)Barycentrics cos A cos(B - C) : cos B cos(C - A) : cos C cos(A - B)
X(343) lies on these lines:
2,6 3,68 5,51 22,161 53,311 76,297 140,569 315,458 427,511 470,634 471,633 472,621 473,622
X(343) = isotomic conjugate of X(275)
X(343) = complement of X(1993)
X(343) = X(I)-Ceva conjugate of X(J) for these (I,J): (76,311), (311,5)
X(343) = X(216)-cross conjugate of X(5)
X(343) = crosspoint of X(69) and X(76)
X(343) = crosssum of X(I) and X(J) for these (I,J): (6,571), (25,32)
X(344) = ISOTOMIC CONJUGATE OF X(277)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (csc2A/2)[cos4(B/2) + cos4(C/2) - cos4(A/2)]
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)
X(344) lies on these lines:
2,37 7,190 8,480 9,69 44,193 45,141 144,320 264,281 319,391
X(344) = isotomic conjugate of X(277)
X(345) = ISOTOMIC CONJUGATE OF X(278)
Trilinears (csc A)/(1 - sec A) : (csc B)/(1 - sec B) : (csc C)/(1 - sec C)= bc(b + c - a)(b2 + c2 - a2) : ca(c + a - b)(c2 + a2 - b2) : ab(a + b - c)(a2 + b2 - c2)
Barycentrics 1/(1 - sec A) : 1/(1 - sec B) : 1/(1 - sec C)
X(345) lies on these lines:
2,37 8,21 22,100 57,728 63,69 78,1040 190,329 219,332 280,341 304,348 498,1089
X(345) = isogonal conjugate of X(608)
X(345) = isotomic conjugate of X(278)
X(345) = X(I)-Ceva conjugate of X(J) for these (I,J): (304,69), (332,78)
X(345) = X(I)-cross conjugate of X(J) for these (I,J): (78,69), (219,8), (306,312)
X(346) = ISOTOMIC CONJUGATE OF X(279)
Trilinears bc(b + c - a)2 : ca(c + a - b)2 : ab(a + b - c)2= cos(A/2) csc3(A/2) : cos(B/2) csc3(B/2) : cos(C/2) csc3(C/2)
Barycentrics (b + c - a)2 : (c + a - b)2 : (a + b - c)2
X(346) lies on these lines:
2,37 6,145 8,9 45,594 69,144 78,280 100,198 219,644 220,1043 253,306 279,304 281,318 573,1018
X(346) = isogonal conjugate of X(1407)
X(346) = isotomic conjugate of X(279)
X(346) = X(312)-Ceva conjugate of X(8)
X(346) = X(200)-cross conjugate of X(8)
X(346) = crosspoint of X(312) and X(341)
X(346) = crosssum of X(604) and X(1106)
X(347) = ISOTOMIC CONJUGATE OF X(280)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (-1 - cos A + cos B + cos C) sec2(A/2)Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
X(347) lies on these lines:
1,7 2,92 8,253 34,452 37,948 69,664 75,280 144,219 223,329 227,322 241,1108 573,1020
X(347) = isogonal conjugate of X(2192)
X(347) = isotomic conjugate of X(280)
X(347) = anticomplement of X(281)
X(347) = X(I)-Ceva conjugate of X(J) for these (I,J): (75,7), (348,2)
X(347) = cevapoint of X(40) and X(223)
X(347) = X(I)-cross conjugate of X(J) for these (I,J): (40,329), (221,196), (227,223)
X(347) = crosspoint of X(75) and X(322)
X(348) = ISOTOMIC CONJUGATE OF X(281)
Trilinears cot A sec2(A/2) : cot B sec2(B/2) : cot C sec2(C/2)= (csc A)/(1 + sec A) : (csc B)/(1 + sec B) : (csc C)/(1 + sec C)
Barycentrics 1/(1 + sec A) : 1/(1 + sec B) : 1/(1 + sec C)
X(348) lies on these lines: 2,85 7,21 8,664 69,73 &n