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Long ago, someone drew a triangle and three segments across it, each starting at a vertex and stopping at the midpoint of the opposite side. The segments met in a point. The person was impressed and repeated the experiment on a different shape of triangle. Again the segments met in a point. The person drew yet a third triangle, very carefully, with the same result. He told his friends. To their surprise and delight, the coincidence worked for them, too. Word spread, and the magic of the three segments was regarded as the work of a higher power.

Centuries passed, and someone proved that the three medians do indeed concur in a point, now called the centroid. The ancients found other points, too, now called the incenter, circumcenter and orthocenter. More centuries passed, more special points were discovered, and a definition of triangle center emerged. Like the definition of continuous function, this definition is satisfied by infinitely many objects, of which only finitely many will ever be published. The Encyclopedia of Triangle Centers (ETC) extends a list of 400 triangle centers published in the book Triangle Centers and Central Triangles. A highly recommended introduction to triangle centers and related geometry is Paul Yiu's A Tour of Triangle Geometry.

Eric Weisstein's MathWorld, the web's most extensive mathematics resource, covers much of classical and modern triangle geometry, including sketches and references. A good place to start is MathWorld's PlaneGeometry. MathWorld is hosted by Wolfram Research, makers of Mathematica. A highly recommended Mathematica package for exploring triangle geometry is described at PlaneGeometry.


NOTATION AND COORDINATES


The reference triangle is ABC, with sidelengths a,b,c. Each triangle center P has homogeneous trilinear coordinates, or simply trilinears, of the form x : y : z. This means there is a nonzero function h of (a,b,c) such that

x = hx', y = hy', z = hz',

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

u = ku', v = kv', w = kw',

where u', v', w' denote the signed areas of triangles PBC, PCA, PAB. Both coordinate systems are widely used; if trilinears for a point are x : y : z, then barycentrics are ax : by : cz.

In order that every center should have its own name, in cases where no particular name arises from geometrical or historical considerations, the name of a star is used instead. For example, X(770) is POINT ACAMAR. For a list of star names, visit SkyEye - (Un)Common Star Names.


HOW TO USE ETC


You won't have to scroll down very far to find well known centers. Other named centers can be found using your computer's searcher - for example, search for "Apollonius" to find "Apollonius point" as X(181).

To determine if a possibly new center is already listed, click Search at the top of this page. If you're unsure of a term, click Glossary. For visual constructions of selected centers with text, click Sketches. To learn about the triangle geometry interest group, Hyacinthos, and other resources, or to view acknowledgments or supplementary encyclopedic material, click Links, Thanks, or More.

If you have The Geometer's Sketchpad, you can view sketches of many of the triangle centers. These are dynamic sketches, meaning that you can vary the shape of the reference triangle A, B, C by dragging these vertices. (For information on Sketchpad, click Sketchpad.) The sketches are also useful for making your own Sketchpad tools, so that you can quickly construct X-of-T for many choices of X and T. For example, starting with ABC and point P, you could efficiently construct center X of the four triangles ABC, BCP, CAP, ABP.


X(1) = INCENTER

Trilinears       1 : 1 : 1
Barycentrics  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

r = 2*area(ABC)/(a + b + c).

Three more points are also equidistant from the sidelines; they are given by these names and trilinears:

A-excenter = -1 : 1 : 1,     B-excenter = 1 : -1 : 1,     C-excenter = 1 : 1 : -1.

The radii of the excircles are

2*area(ABC)/(-a + b + c), 2*area(ABC)/(a - b + c), 2*area(ABC)/(a + b - c).

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,

area(ABC) = sqrt(r*ra*rb*rc) and ra + rb + rc = r + 4R,

where R denotes the radius of the circumcircle.

The incenter is the identity of the group of triangle centers under "trilinear multiplication" defined by

(x : y : z)*(u : v : w) = xu : yv : zw.

A construction for * is easily obtained from the construction for "barycentric multiplication" mentioned in connection with X(2), just below.

The incenter and the other classical centers are discussed in these highly recommended books:

Paul Yiu, Introduction to the Geometry of the Triangle, 2002;
Nathan Altshiller Court, College Geometry, Barnes & Noble, 1952;
Roger A. Johnson, Advanced Euclidean Geometry, Dover, 1960.

X(1) lies on these lines:
2,8   3,35   4,33   5,11   6,9   7,20   15,1251   16,1250   19,28   21,31   24,1061   25,1036   29,92   30,79   32,172   39,291   41,101   49,215   60,110   61,203   62,202   64,1439   69,1245   71,579  75,86   76,350   82,560   84,221   87,192   88,100   90,155   99,741   102,108   104,109   142,277   147,150   159,1486   163,293   164,258   166,1488   167,174   168,173   179,1142   181,970   182,983   184,1726   185,296   188,361   190,537   196,207   201,212   204,1712   224,377   227,1465   228,1730   229,267   256,511   257,385   280,1256 281,282   289,363   312,1089   318,1897   320,752   321,964   329,452   335,384   336,811   341,1050   344,1265   346,1219   357,1508   358,1507   364,365   371,1702   372,1703   376,553   378,1063   393,836   394,1711   409,1247   410,1248   411,1254   442,1834   474,1339   475,1861   512,875   513,764   514,663   522,1459   528,1086   561,718   563,1820   564,1048   572,604   573,941   574,1571   594,1224   607,949   631,1000   644,1280   647,1021   650,1643   651,1156   659,891   662,897   672,1002   689,719   704,1502   727,932   731,789   748,756   761,825   765,1052   810,1577   840,1308   905,1734   908,998   921,1800   939,1260   945,1875   947,1753   951,1435   969,1444   971,1419   989,1397   1013,1430   1037,1041   1053,1110   1057,1598   1059,1597   1075,1148   1106,1476   1168,1318   1170,1253   1185,1206   1197,1613   1292,1477   1333,1761   1342,1700   1343,1701   1361,1364   1389,1393   1399,1727   1406,1480   1409,1765   1437,1710   1472,1791   1719,1790   1855,1886   1859,1871   1872,1887

X(1) is the {X(2),X(8)}-harmonic conjugate of X(10). For a list of other harmonic conjugates of X(1), click More at the top of this page.

X(1) = midpoint of X(I) and X(J) for these (I,J): (7,390), (8,145)

X(1) = reflection of X(I) in X(J) for these (I,J): (2,551), (3,1385), (4,946), (6,1386), (8,10), (9,1001), (10,1125), (11,1387), (36,1319), (40,3), (43,995), (46,56), (57,999), (63,993), (65,942), (72,960), (80,11), (100,214), (191,21), (200,997), (238,1279), (267,229), (291,1015), (355,5), (484,36), (984,37), (1046,58), (1054,106), (1478,226)

X(1) = isogonal conjugate of X(1)
X(1) = isotomic conjugate of X(75)
X(1) = cyclocevian conjugate of X(1029)
X(1) = inverse-in-circumcircle of X(36)
X(1) = inverse-in-Fuhrmann-circle of X(80)
X(1) = inverse-in-Bevan-circle of X(484)
X(1) = complement of X(8)
X(1) = anticomplement of X(10)
X(1) = anticomplementary conjugate of X(1330)
X(1) = complementary conjugate at X(1329)
X(1) = eigencenter of cevian triangle of X(I) for I = 1, 88, 162
X(1) = eigencenter of anticevian triangle of X(I) for I = 1, 44, 513

X(1) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,9), (4,46), (6,43), (7,57), (8,40), (9,165), (10,191), (21,3), (29,4), (75,63), (77,223), (78,1490), (80,484), (81,6), (82,31), (85,169), (86,2), (88,44), (92,19), (100,513), (104,36), (105,238), (174,173), (188,164), (220,170), (259,503), (266,361), (280,84), (366,364), (508,362), (1492,1491)

X(1) = cevapoint of X(I) and X(J) for these (I,J):
(2,192), (6,55), (11,523), (15,202), (16,203), (19,204), (31,48), (34,207), (37,42), (50,215), (56,221), (65,73), (78,1490), (244,513)

X(1) = X(I)-cross conjugate of X(J) for these (I,J):
(2,87), (3,90), (6,57), (31,19), (33,282), (37,2), (38,75), (42,6), (44,88), (48,63), (55,9), (56,84), (58,267), (65,4), (73,3), (192,43), (207,1490), (221,40), (244,513), (259,258), (266,505), (354,7), (367,366), (500,35), (513,100), (517,80), (518,291), (1491,1492)

X(1) = crosspoint of X(I) and X(J) for these (I,J):
(2,7), (8,280), (21,29), (59,110), (75,92), (81,86)

X(1) = crosssum of X(I) and X(J) for these (I,J):
(2,192), (4,1148), (6,55), (11,523), (15,202), (16,203), (19,204), (31,48), (34,207), (37,42), (44,678), (50,215), (56,221), (57,1419), (65,73), (207,1490), (214,758), (244,513), (500,942), (512,1015), (774,820), (999,1480)

X(1) = crossdifference of any two points on line X(44)X(513)

X(1) = X(I)-Hirst inverse of X(J) for these (I,J):
(2,239), (4,243), (6,238), (9,518), (19,240), (57,241), (105,294), (291,292).

X(1) = X(6)-line conjugate of X(44)

X(1) = X(I)-aleph conjugate of X(J) for these (I,J):
(1,1), (2,63), (4,920), (21,411), (29,412), (88,88), (100,100), (162,162), (174,57), (188,40), (190,190), (266,978), (365,43), (366,9), (507,173), (508,169), (513,1052), (651, 651), (653,653), (655,655), (658,658), (660,660), (662,662), (673,673), (771,771), (799,799), (823,823), (897,897)

X(1) = X(I)-beth conjugate of X(J) for these (I,J):
(1,56), (2,948), (8,8), (9,45), (21,1), (29,34), (55,869), (99,85), (100,1), (110,603), (162,208), (643,1), (644,1), (663,875), (664,1), (1043,78)


X(2) = CENTROID

Trilinears       1/a : 1/b : 1/c
                        = bc : ca : ab
                        = csc A : csc B : csc C
                        = cos A + cos B cos C : cos B + cos C cos A : cos C + cos A cos B
                        = sec A + sec B sec C : sec B + sec C sec A : sec C + sec A sec B
                        = cos A + cos(B - C) : cos B + cos(C - A) : cos C + cos(A - B)
                        = cos B cos C - cos(B - C) : cos C cos A - cos(C - A) : cos A cos B - cos(A - B)

Barycentrics  1 : 1 : 1

X(2) is the point of concurrence of the medians of ABC, situated 1/3 of the distance from each vertex to the midpoint of the opposite side. More generally, if L is any line in the plane of ABC, then the distance from X(2) to L is the average of the distances from A, B, C to L. An idealized triangular sheet balances atop a pin head located at X(2) and also balances atop any knife-edge that passes through X(2). The triangles BXC, CXA, AXB have equal areas if and only if X = X(2).

If you have The Geometer's Sketchpad, you can view Centroid.

X(2) is the centroid of the set of 3 vertices, the centroid of the triangle including its interior, but not the centroid of the triangle without its interior; that centroid is X(10).

X(2) is the identity of the group of triangle centers under "barycentric multiplication" defined by

(x : y : z)*(u : v : w) = xu : yv : zw.

X(2) lies on these lines:
1,8   3,4   6,69   7,9   11,55   12,56   13,16   14,15   17,62   18,61   19,534   31,171   32,83   33,1040   34,1038   35,1479   36,535  37,75   38,244   39,76   40,946   44,89   45,88   51,262   52,1216   54,68   58,540   65,959   66,206   71,1246   72,942   74,113   77,189   80,214   85,241   92,273   94,300   95,97   98,110   99,111   101,116   102,117   103,118   104,119   106,121   107,122   108,123   109,124   112,127   128,1141   129,1298   130,1303   131,1300   133,1294   136,925   137,930   154,1503   165,516   169,1763   174,236   176,1659   178,188   187,316   196,653   201,1393   210,354   216,232   220,1170   222,651   231,1273   242,1851   243,1857   252,1166   253,1073   254,847   257,1432   261,593   265,1511   271,1034   272,284   280,318   283,580   290,327   292,334   294,949   308,702   311,570   314,941   319,1100   322,1108   330,1107   341,1219   351,804   355,944   360,1115   366,367   371,486   372,485   392,517   476,842   480,1223   489,1132   490,1131   495,956   496,1058   514,1022   523,1649   525,1640   561,716   568,1154   572,1746   573,1730   578,1092   585,1336   586,1123   588,1504   589,1505   594,1255   647,850   648,1494   650,693   664,1121   668,1015   670,1084   689,733   743,789   799,873   812,1635   846,1054   914,1442   918,1638   927,1566   954,1260   968,1738   1000,1145   1043,1834   1060,1870   1074,1785   1076,1838   1089,1224   1093,1217   1124,1378   1143,1489   1155,1836   1171,1509   1186,1207   1257,1265   1284,1403   1335,1377   1340,1349   1341,1348   1500,1574   1501,1691   1672,1681   1673,1680   1674,1679   1675,1678   1697,1706

X(2) is the {X(3),X(5)}-harmonic conjugate of X(4). For a list of other harmonic conjugates of X(2), click More at the top of this page.

X(2) = midpoint of X(I) and X(J) for these (I,J): (3,381), (4,376), (210,354)
X(2) = reflection of X(I) in X(J) for these (I,J): (1,551), (3,549), (4,381), (5,547), (6,597), (20,376), (69,599), (148,671), (376,3), (381,5), (549,140), (551,1125), (599,141), (671,115), (903,1086), (1121,1146)

X(2) = isogonal conjugate of X(6)
X(2) = isotomic conjugate of X(2)
X(2) = cyclocevian conjugate of X(4)
X(2) = inverse-in-circumcircle of X(23)
X(2) = inverse-in-nine-point-circle of X(858)
X(2) = inverse-in-Brocard-circle of X(110)
X(2) = complement of X(2)
X(2) = anticomplement of X(2)
X(2) = anticomplementary conjugate of X(69)
X(2) = complementary conjugate of X(141)

X(2) = X(I)-Ceva conjugate of X(J) for these (I,J):
(1,192), (4,193), (6,194), (7,145), (8,144), (30,1494), (69,20), (75,8), (76,69), (83,6), (85,7), (86,1), (87,330), (95,3), (98,385), (99,523), (190,514), (264,4), (274,75), (276, 264), (287,401), (290,511), (308,76), (312,329), (325,147), (333,63), (348,347), (491,487), (492,488), (523,148), (626,1502)

X(2) = cevapoint of X(I) and X(J) for these (I,J):
(1,9), (3,6), (5,216), (10,37), (32,206), (39,141), (44,214), (57,223), (114,230), (115,523), (128,231), (132,232), (140,233), (188,236)

X(2) = X(I)-cross conjugate of X(J) for these (I,J):
(1,7), (3,69), (4,253), (5,264), (6,4), (9,8), (10,75), (32,66), (37,1), (39,6), (44,80), (57,189), (75,330), (114,325), (140,95), (141,76), (142,85), (178,508), (187,67), (206,315), (214,320), (216,3), (223,329), (226,92), (230,98), (233,5), (281,280), (395,14), (396,13), (440,306), (511,290), (514,190), (523,99)

X(2) = crosspoint of X(I) and X(J) for these (I,J):
(1,87), (75,85), (76,264), (83,308), (86,274), (95,276)

X(2) = crosssum of X(I) and X(J) for these (I,J):
(1,43), (2,194), (31,41), (32,184), (42,213), (51,217), (125,826), (649,1015), (688,1084), (902,1017), (1400,1409)

X(2) = crossdifference of any two points on line X(187)X(237)

X(2) = X(I)-Hirst inverse of X(J) for these (I,J):
(1,239), (3,401), (4,297), (6,385), (21,448), (27,447), (69,325), (75,350), (98,287), (115,148), (193,230), (291,335), (298,299), (449,452)

X(2) = X(3)-line conjugate of X(237) = X(316)-line conjugate of X(187)

X(2) = X(I)-aleph conjugate of X(J) for these (I,J):
(1,1045), (2,191), (86,2), (174,1046), (333,20), (366,846)

X(2) = X(I)-beth conjugate of X(J) for these (I,J):
(2,57), (21,995) (190,2), (312,312), (333,2), (643,55), (645,2), (646,2), (648,196), (662,222)

X(2) is the unique point X (as a function of a,b,c) for which the vector-sum XA + XB + XC is the zero vector.


X(3) = CIRCUMCENTER

Trilinears       cos A : cos B : cos C
                        = a(b2 + c2 - a2) : b(c2 + a2 - b2) : c(a2 + b2 - c2)

Barycentrics  sin 2A : sin 2B : sin 2C

X(3) is the point of concurrence of the perpendicular bisectors of the sides of ABC. The lengths of segments AX, BX, CX are equal if and only if X = X(3). This common distance is the radius of the circumcircle, which passes through vertices A,B,C. Called the circumradius, it is given by

R = a/(2 sin A) = abc/(4*area(ABC)).

If you have The Geometer's Sketchpad, you can view Circumcenter.

X(3) lies on these lines:
1,35   2,4   6,15   7,943   8,100   9,84   10,197   11,499   12,498   13,17   14,18   19,1871   31,601   33,1753   34,1465   37,975   38,976   41,218   42,967   47,1399   48,71   49,155   54,97   60,1175   63,72   64,154   66,141   67,542  68,343   69,332   73,212   74,110   76,98   77,1410   83,262   86,1246   90,1898   95,264   101,103   102,109   105,277   106,1293   107,1294   108,1295   111,1296   112,1297   113,122   114,127   119,123   125,131   128,1601   142,516   143,1173   145,1483   149,1484   158,243   161,1209   169,910   191,1768   193,1353   194,385   200,963   201,1807   207,1767   223,1035   225,1074   227,1455   238,978   252,930   256,987   269,939   296,820   298,617   299,616   302,621   303,622   305,1799   315,325   345,1791   347,1119   348,1565   352,353   388,495   390,1058   393,1217   395,398   396,397   476,477   485,590   486,615   489,492   490,491   496,497   525,878   595,995   611,1469   613,1428   618,635   619,636   623,629   624,630   639,641   640,642   653,1148   662,1098   667,1083   691,842   695,1613   847,925   901,953   902,1201   917,1305   920,1858   934,972   945,1457   950,1210   951,1407   955,1170   960,997   962,1621   1000,1476   1033,1249   1037,1066   1054,1283   1055,1334   1057,1450   1093,1105   1167,1413   1177,1576   1180,1627   1184,1194   1196,1611   1298,1303   1331,1797   1364,1795   1397,1682   1398,1870   1406,1464   1411,1772   1427,1448   1452,1905   1728,1864   1737,1837   1770,1836   1779,1780

X(3) is the {X(2),X(4)}-harmonic conjugate of X(5). For a list of other harmonic conjugates of X(3), click More at the top of this page. If triangle ABC is acute, then X(3) is the incenter of the tangential triangle and the Bevan point, X(40), of the orthic triangle.

X(3) = midpoint of X(I) and X(J) for these (I,J):
(1,40), (2,376), (4,20), (22,378), (74,110), (98,99), (100,104), (101,103), (102,109), (476,477)

X(3) = reflection of X(I) in X(J) for these (I,J):
(1,1385), (2,549), (4,5), (5,140), (6,182), (20,550), (52,389), (110,1511), (114,620), (145,1483), (149,1484), (155,1147), (193,1353), (195,54), (265,125), (355,10), (381,2), (382,4), (399,110), (550,548), (576,575), (946,1125), (1351,6), (1352,141), (1482,1)

X(3) = isogonal conjugate of X(4)
X(3) = isotomic conjugate of X(264)
X(3) = inverse-in-nine-point-circle of X(2072)
X(3) = inverse-in-orthocentroidal-circle of X(5)
X(3) = inverse-in-1st-Lemoine-circle of X(2456)
X(3) = inverse-in-2nd-Lemoine-circle of X(1570)
X(3) = complement of X(4)
X(3) = anticomplement of X(5)
X(3) = complementary conjugate of X(5)
X(3) = eigencenter of the medial triangle
X(3) = eigencenter of the tangential triangle

X(3) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,6), (4,155), (5,195), (20,1498), (21,1), (22,159), (30,399), (63,219), (69,394), (77,222), (95,2), (96,68), (99,525), (100,521), (110,520), (250, 110), (283,255)

X(3) = cevapoint of X(I) and X(J) for these (I,J):
(6,154), (48,212), (55,198), (71,228), (185,417), (216,418)

X(3) = X(I)-cross conjugate of X(J) for these (I,J):
(48,222), (55,268), (71,63), (73,1), (184,6), (185,4), (212,219), (216,2), (228,48), (520,110)

X(3) = crosspoint of X(I) and X(J) for these (I,J):
(1,90), (2,69), (4,254), (21,283), (54,96), (59,100), (63,77), (78,271), (81,272), (95,97), (99,249), (110,250), (485,486)

X(3) = crosssum of X(I) and X(J) for these (I,J):
(1,46), (2,193), (3,155), (5,52), (6,25), (11,513), (19,33), (30,113), (37,209), (39, 211), (51,53), (65,225), (114,511), (115,512), (116,514), (117, 515), (118,516), (119,517), (120,518), (121,519), (122,520), (123,521), (124,522), (125,523), (126,524), (127,525), (128,1154), (136,924), (184,571), (185,235), (371,372), (487,488)

X(3) = crossdifference of any two points on line X(230)X(231)
X(3) = X(I)-Hirst inverse of X(J) for these (I,J): (2, 401), (4,450), (6,511), (21,416), (194, 385)
X(3) = X(2)-line conjugate of X(468)

X(3) = X(I)-aleph conjugate of X(J) for these (I,J): (1,1046), (21,3), (188,191), (259,1045)

X(3) = X(I)-beth conjugate of X(J) for these (I,J):
(3,603), (8,355), (21,56), (78,78), (100,3), (110,221), (271,84), (283,3), (333,379), (643,8)


X(4) = ORTHOCENTER

Trilinears       sec A : sec B : sec C
                        = cos A - sin B sin C : cos B - sin C sin A : cos C - sin A sinB
                        = cos A - cos(B - C) : cos B - cos(C - A) : cos C - cos(A - B)
                        = sin B sin C - cos(B - C) : sin C sin A - cos(C - A) : sin A sin B - cos(A - B)

Barycentrics  tan A : tan B : tan C

X(4) is the point of concurrence of the altitudes of ABC.

If you have The Geometer's Sketchpad, you can view Orthocenter.

X(4) and the vertices A,B,C comprise an orthocentric system, defined as a set of four points, one of which is the orthocenter of the triangle of the other three (so that each is the orthocenter of the other three). The incenter and excenters also form an orthocentric system. If ABC is acute, then X(4) is the incenter of the orthic triangle.

Suppose P is not on a sideline of ABC. Let A'B'C' be the cevian triangle of P. Let D,E,F be the circles having diameters AA', BB',CC'. The radical center of D,E,F is X(4). (Floor van Lamoen, Hyacinthos #214, Jan. 24, 2000.)

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 2: The Orthocenter.

X(4) lies on these lines:
1,33   2,3   6,53   7,273   8,72   9,10   11,56   12,55   13,61   14,62   15,17   16,18   32,98   35,498   36,499   37,1841   39,232   42,1860   46,90   48,1881   49,156   51,185   52,68   54,184   57,84   64,1853   65,158   67,338   69,76   74,107   78,908   79,1784   80,1825   83,182   93,562  94,143   96,231   99,114   100,119   101,118   102,124   103,116   109,117   110,113   111,1560   120,1292   121,1293   122,1294   123,1295   126,1296   127,1289   128,930   129,1303   130,1298   131,135   137,933   141,1350   145,149   147,148   150,152   155,254   162,270   165,1698   171,601   193,1351   195,399   204,1453   218,294   238,602   240,256   250,1553   252,1487   276,327   279,1565   371,485   372,486   390,495   394,1217   477,1304   487,489   488,490   496,999   512,879   523,1552   542,576   569,1179   572,1474   574,1506   575,598   579,1713   580,1714   590,1151   608,1518   615,1152   616,627   617,628   653,1156   774,1254   801,1092   842,935   937,1534   940,1396   941,1880   953,1309   1036,1065   1037,1067   1038,1076   1039,1096   1040,1074   1160,1162   1161,1163   1251,1832   1329,1376   1340,1348   1341,1349   1385,1538   1430,1468   1499,1550   1715,1730   1716,1721   1717,1718   1726,1782

X(4) is the {X(3),X(5)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(4), click More at the top of this page.

X(4) = midpoint of X(I) and X(J) for these (I,J):
(3,382), (147,148), (149,153), (150,152)

X(4) = reflection of X(I) in X(J) for these (I,J): (1,946), (2,381), (3,5), (5,546), (8,355), (20,3), (24,235), (25,1596), (40,10), (69,1352), (74,125), (98,115), (99,114), (100,119), (101,118), (102,124), (103,116), (104,11), (107,133), (109,117), (110,113), (112,132), (145,1482), (185,389), (186,403), (193,1351), (376,2), (378,427), (550,140), (925,131), (930,128), (944,1), (1113,1312), (1114,1313), (1141,137), (1292,120), (1293,121), (1294,122), (1295,123), (1296,126), (1297,127), (1298,130), (1299,135), (1300,136), (1303, 129), (1350,141), (1593,1595)

X(4) = isogonal conjugate of X(3)
X(4) = isotomic conjugate of X(69)
X(4) = cyclocevian conjugate of X(2)
X(4) = inverse-in-circumcircle of X(186)
X(4) = inverse-in-nine-point-circle of X(403)
X(4) = complement of X(20)
X(4) = anticomplement of X(3)
X(4) = anticomplementary conjugate of X(20)
X(4) = eigencenter of cevian triangle of X(I) for I = 1, 88, 162
X(4) = eigencenter of anticevian triangle of X(I) for I = 1, 44, 513

X(4) = X(I)-Ceva conjugate of X(J) for these (I,J):
(7,196), (27,19), (29,1), (92,281), (107,523), (264,2), (273,278), (275,6), (286,92), (393, 459)

X(4) = cevapoint of X(I) and X(J) for these (I,J):
(1,46), (2,193), (3,155), (5,52), (6,25), (11,513), (19,33), (30,113), (34,208), (37,209), (39,211), (51,53), (65,225), (114,511), (115,512), (116,514), (117,515), (118,516), (119,517), (120,518), (121,519), (122,520), (123,521), (124,522), (125,523), (126,524), (127,525), (185,235), (371,372), (487,488)

X(4) = X(I)-cross conjugate of X(J) for these (I,J):
(3,254), (6,2), (19,278), (25,393), (33,281), (51,6), (52,24), (65,1), (113,403), (125,523), (185,3), (193,459), (225,158), (389,54), (397,17), (398,18), (407,225), (427,264), (512,112), (513,108), (523,107)

X(4) = crosspoint of X(I) and X(J) for these (I,J): (2,253), (7,189), (27,286), (92,273)

X(4) = crosssum of X(I) and X(J) for these (I,J):
(4,1075), (6,154), (25,1033), (48,212), (55,198), (56,1035), (71,228), (184,577), (185,417), (216,418)

X(4) = crossdifference of any two points on line X(520)X(647)

X(4) = X(I)-Hirst inverse of X(J) for these (I,J):
(1,243), (2,297), (3,350), (19,242), (21,425), (24,421), (25,419), (27,423), (28,422), (29,415), (420,427), (424,451), (459,460), (470,471), (1249,1503)

X(4) = X(I)-aleph conjugate of X(J) for these (I,J): (1,1047), (29,4)

X(4) = X(I)-beth conjugate of X(J) for these (I,J):
(4,34), (8,40), (29,4), (162,56), (318,318), (811,331)


X(5) = NINE-POINT CENTER

Trilinears       cos(B - C) : cos(C - A) : cos(A - B)
                        = f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A + 2 cos B cos C
                        = g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = cos A - 2 sin B sin C
                        = h(a,b,c) : h(b,c,a): h(c,a,b), where h(a,b,c) = bc[a2(b2 + c2) - (b2 - c2)2]

Barycentrics  a cos(B - C) : b cos(C - A) : c cos(A - B)
                        = h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = a2(b2 + c2) - (b2 - c2)2

X(5) is the center of the nine-point circle. Euler showed in that this circle passes through the midpoints of the sides of ABC and the feet of the altitudes of ABC, hence six of the nine points. The other three are the midpoints of segments A-to-X(4), B-to-X(4), C-to-X(4). The radius of the nine-point circle is one-half the circumradius.

Dan Pedoe, Circles: A Mathematical View, Mathematical Association of America, 1995.

If you have The Geometer's Sketchpad, you can view Nine-point center and Euler Line.

X(5) lies on these lines:
1,11   2,3   6,68   8,1389   9,1729   10,517   13,18   14,17   15,2913   16,2912   32,230   33,1062   34,1060   39,114   40,1698   46,1836   49,54   51,52   53,216   55,498   56,499   57,1728   65,1737   69,1351   72,908   76,262   79,1749   83,98   85,1565   96,1166   113,125   116,118   117,124   122,133   127,132   128,137   129,130   131,136   141,211   142,971   156,184   182,206   183,315   217,1625   225,1465   226,912   252,1157   264,1093   298,634   299,633   302,622   303,621   311,1225   316,1078   339,1235   371,590   372,615   386,1834   388,999   392,1512   491,637   492,638   515,1125   524,576   539,1493   542,575   570,1879   573,1213   578,1147   579,1901   582,1754   601,750   602,748   618,629   619,630   842,1287   920,1454   1073,1217   1090,1091   1155,1770   1173,1487   1181,1899   1214,1838   1498,1853   1848,1871   1861,1872

X(5) is the {X(2),X(4)}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(5), click More at the top of this page.

X(5) = midpoint of X(I) and X(J) for these (I,J):
(1,355), (2,381), (3,4), (11,119), (20,382), (68,155), (110,265), (113,125), (114,115), (116,118), (117,124), (122,133), (127,132), (128,137), (129,130), (131,136)

X(5) = reflection of X(I) in X(J) for these (I,J): (2,547), (3,140), (4,546), (20,548), (52,143), (549,2), (550,3), (1263,137), (1353,6), (1385,1125), (1483,1), (1484,11)

X(5) = isogonal conjugate of X(54)
X(5) = isotomic conjugate of X(95)
X(5) = inverse-in-circumcircle of X(2070)
X(5) = inverse-in-orthocentroidal-circle of X(3)
X(5) = complement of X(3)
X(5) = anticomplement of X(140)
X(5) = complementary conjugate of X(3)
X(5) = eigencenter of anticevian triangle of X(523)

X(5) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,216), (4,52), (110,523), (264, 324), (265,30), (311,343), (324,53)

X(5) = cevapoint of X(I) and X(J) for these (I,J): (3,195), (51,216)

X(5) = X(I)-cross conjugate of X(J) for these (I,J): (51,53), (216,343), (233,2)
X(5) = crosspoint of X(I) and X(J) for these (I,J): (2,264), (311,324)
X(5) = crosssum of X(I) and X(J) for these (I,J): (3,1147), (6,184)
X(5) = crossdifference of any two points on line X(50)X(647)
X(5) = X(1)-aleph conjugate of X(1048)


X(6) = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT)

Trilinears       a : b : c
                        = sin A : sin B : sin C

Barycentrics  a2 : b2 : c2

X(6) is the point of concurrence of the symmedians (reflections of medians in corresponding angle bisectors); the point (x, y, z), given here in actual trilinear distances, that minimizes x2 + y2 + z2.

If you have The Geometer's Sketchpad, you can view Symmedian point.

Let A'B'C' be the pedal triangle of an arbitrary point X, and let S(X) be the vector sum XA' + XB' + XC'. Then

S(X) = (0 vector) if and only if X = X(6).

The "if" implication is equivalent to the well known fact that X(6) is the centroid of its pedal triangle, and the converse was proved by Barry Wolk (Hyacinthos #19, Dec. 23, 1999).

X(6) is the radical trace of the 1st and 2nd Lemoine circles. (Peter J. C. Moses, 8/24/03)

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 7: The Symmedian Point.

X(6) lies on these lines:
1,9   2,69   3,15   4,53   5,68   7,294   8,594   10,1377   13,14   17,18   19,34   21,941   22,251   23,353   24,54   25,51   26,143   27,1246   31,42   33,204   36,609  40,380   41,48   43,87   57,222   60,1169   64,185   66,427   67,125   70,1594   74,112   75,239   76,83   77,241   88,89   98,262   99,729   100,739   101,106   105,1002   110,111   145,346   157,248   160,237   162,1013   169,942   181,197   190,192   194,384   210,612   226,1751   256,1580   264,287   274,1218   279,1170   281,1146   282,1256   291,985   292,869   297,317   305,1241   314,981   330,1258   344,1332   354,374   442,1714   493,1583   494,1584   513,1024   517,998   519,996   523,879   560,1631   561,720   588,1599   589,1600   593,1171   595,1126   598,671   603,1035   644,1120   657,1459   662,757   688,882   689,703   691,843   692,1438   694,1084   706,1502   717,789   750,899   753,825   755,827   840,919   846,1051   893,1403   909,1415   911,1461   939,1802   943,1612   947,1622   959,961   963,1208   967,1790   971,990   986,1046   1096,1859   1112,1177   1131,1132   1139,1140   1166,1601   1173,1614   1174,1617   1195,1399   1201,1696   1214,1708   1327,1328   1362,1416   1399,1425   1423,1429   1718,1781   1826,1837   1836,1839   1854,1858

X(6) is the {X(15),X(16)}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(6), click More at the top of this page.

X(6) = midpoint of X(69) and X(193)
X(6) = reflection of X(I) in X(J) for these (I,J): (1,1386), (2,597), (3,182), (67,125), (69,141), (159,206), (182,575), (592,2), (694,1084), (1350,3), (1351,576), (1352,5)

X(6) = isogonal conjugate of X(2)
X(6) = isotomic conjugate of X(76)
X(6) = cyclocevian conjugate of X(1031)
X(6) = inverse-in-circumcircle of X(187)
X(6) = inverse-in-orthocentroidal-circle of X(115)
X(6) = inverse-in-1st-Lemoine-circle of X(1691)
X(6) = complement of X(69)
X(6) = anticomplement of X(141)
X(6) = anticomplementary conjugate of X(1369)
X(6) = complementary conjugate of X(1368)

X(6) = X(I)-Ceva conjugate of X(J) for these (I,J):
(1,55), (2,3), (3,154), (4,25), (7,1486), (8,197), (9,198), (10,199), (54,184), (57, 56), (58,31), (68,161), (69,159), (74,1495), (76,22), (81,1), (83,2), (88,36), (95,160), (98,237), (110,512), (111,187), (222,221), (249,110), (251,32), (264,157), (275,4), (284,48), (287,1503), (288,54), (323,399), (394,1498)

X(6) = cevapoint of X(I) and X(J) for these (I,J): (1,43), (2,194), (31,41), (32,184), (42,213), (51,217)

X(6) = X(I)-cross conjugate of X(J) for these (I,J):
(25,64), (31,56), (32,25), (39,2), (41,55), (42,1), (51,4), (184,3), (187,111), (213,31), (217,184), (237,98), (512,110)

X(6) = crosspoint of X(I) and X(J) for these (I,J):
(1,57), (2,4), (9,282), (54,275), (58,81), (83, 251), (110,249), (266,289)

X(6) = crosssum of X(I) and X(J) for these (I,J):
(1,9), (3,6), (4,1249), (5,216), (10,37), (11,650), (32,206), (39,141), (44,214), (56,478), (57,223), (114,230), (115,523), (125,647), (128,231), (132,232), (140,233), (142,1212), (188,236), (226,1214), (244,661), (395,619), (396, 618), (512,1084), (513,1015), (514,1086), (522,1146), (570,1209), (577,1147), (590,641), (615,642), (1125,1213), (1196,1368)

X(6) = crossdifference of any two points on line X(30)X(511)

X(6) = X(I)-Hirst inverse of X(J) for these (I,J): (1,238), (2,385), (3,511), (15,16), (25,232), (56,1458), (58,1326), (523,1316), (1423,1429)

X(6) = X(I)-line conjugate of X(J) for these (I,J): (1,518), (2,524), (3,511)
X(6) = X(I)-aleph conjugate of X(J) for these (I,J): (1,846), (81,6), (365,1045), (366,191), (509,1046)

X(6) = X(I)-beth conjugate of X(J) for these (I,J):
(6,604), (9,9), (21,1001), (101,6), (284,6), (294,6), (644,6), (645,6), (651,6), (652,7), (666,6)


X(7) = GERGONNE POINT

Trilinears       bc/(b + c - a) : ca/(c + a - b) : ab/(a + b - c)
                        = sec2(A/2) : sec2(B/2) : sec2(C/2)

Barycentrics  1/(b + c - a) : 1/(c + a - b) : 1/(a + b - c)

Let A', B', C' denote the points in which the incircle meets the sides BC, CA, AB, respectively. The lines AA', BB', CC' concur in X(7).

If you have The Geometer's Sketchpad, you can view Gergonne point.

X(7) lies on these lines:
1,20   2,9   3,943   4,273   6,294   8,65   11,658   12,1268   21,56   27,81   37,241   33,1041   34,1039   58,272   59,1275   72,443   73,1246   76,1479   80,150   92,189   100,1004   104,934   108,1013   109,675   145,1266   171,983   174,234   177,555   190,344   192,335   193,239   218,277   220,1223   225,969   238,1471   253,280   256,982   274,959   281,653   286,331   310,314   330,1432   349,1269   354,479   404,1259   452,1467   464,1214   480,1376   492,1267   513,885   517,1000   528,664   554,1082   594,599   604,1429   757,1414   840,927   857,1901   870,1431   940,1407   941,1427   944,1389   952,1159   986,1254   987,1106   1002,1362   1020,1765   1061,1870   1354,1367   1365,1366   1386,1456   1419,1449   1435,1848   1486,1602   1617,1621

X(7) is the {X(69),X(75)}-harmonic conjugate of X(8). For a list of other harmonic conjugates of X(7), click More at the top of this page.

X(7) = reflection of X(I) in X(J) for these (I,J): (9,142), (144,9), (390,1), (673,1086), (1156,11)
X(7) = isogonal conjugate of X(55)
X(7) = isotomic conjugate of X(8)
X(7) = cyclocevian conjugate of X(7)
X(7) = inverse-in-incircle of (1323)
X(7) = complement of X(144)
X(7) = anticomplement of X(9)
X(7) = anticomplementary conjugate of X(329)
X(7) = X(I)-Ceva conjugate of X(J) for these (I,J): (75,347), (85,2), (86,77), (286,273), (331,278)

X(7) = cevapoint of X(I) and X(J) for these (I,J):
(1,57), (2,145), (4,196), (11,514), (55,218), (56,222), (63,224), (65,226), (81,229), (177,234)

X(7) = X(I)-cross conjugate of X(J) for these (I,J):
(1,2), (4,189), (11,514), (55,277), (56,278), (57,279), (65,57), (177,174), (222,348), (226,85), (354,1), (497,8), (517,88)

X(7) = crosspoint of X(I) and X(J) for these (I,J): (75,309), (86,286)
X(7) = crosssum of X(I) and X(J) for these (I,J): (41,1253), (42,228)
X(7) = crossdifference of any two points on line X(657)X(663)
X(7) = X(57)-Hirst inverse of X(1447)

X(7) = X(I)-beth conjugate of X(J) for these (I,J):
(7,269), (21,991), (75,75), (86,7), (99,7), (190,7), (290,7), (314,69), (645,344), (648,7), (662,6), (664,7), (666,7), (668,7), (670,7), (671,7), (811,264), (886,7), (889,7), (892,7), (903,7)


X(8) = NAGEL POINT

Trilinears       (b + c - a)/a : (c + a - b)/b : (a + b - c)/c
                        = csc2(A/2) : csc2(B/2) : csc2(C/2)

Barycentrics  b + c - a : c + a - b : a + b - c

Let A'B'C' be the points in which the A-excircle meets BC, the B-excircle meets CA, and the C-excircle meets AB, respectively. The lines AA', BB', CC' concur in X(8). Another construction of A' is to start at A and trace around ABC half its perimeter, and similarly for B' and C'. Also, X(8) is the incenter of the anticomplementary triangle.

If you have The Geometer's Sketchpad, you can view Nagel point.

X(8) lies on these lines:
1,2   3,100   4,72   5,1389   6,594   7,65   9,346   11,1320   19,1891   20,40   21,55   29,219   31,987   33,1039   34,1041   35,993   37,941   38,986   56,404   57,1219   58,996   76,668   79,758   80,149   81,1010   101,1311   140,1483   144,516   171,1468   175,1270   176,1271   177,556   178,236   181,959   190,528   192,256   193,894   194,730   197,1603   210,312   213,981   220,294   221,651   224,914   238,983   253,307   274,1002   277,1280   278,1257   291,330   314,1264   315,760   326,1442   344,480   348,664   392,1000   405,943   406,1061   442,495   443,942   474,999   475,1063   491,1267   595,1724   599,1086   631,1385   643,1098   726,1278   860,1068   908,946   961,1460   1015,1574   1016,1083   1034,1895   1036,1183   1124,1377   1211,1834   1281,1282   1317,1388   1335,1378   1500,1573   1672,1680   1673,1681   1674,1679   1675,1679   1857,1896

X(8) is the {X(69),X(75)}-harmonic conjugate of X(7). For a list of other harmonic conjugates of X(8), click More at the top of this page.

X(8) = reflection of X(I) in X(J) for these (I,J): (1,10), (4,355), (20,40), (100,1145), (145,1), (149,80), (192,984), (390,9), (944,2), (962,4), (1320,11), (1482,5), (1483,140)

X(8) = isogonal conjugate of X(56)
X(8) = isotomic conjugate of X(7)
X(8) = cyclocevian conjugate of X(189)
X(8) = complement of X(145)
X(8) = anticomplement of X(1)
X(8) = anticomplementary conjugate of X(8)
X(8) = X(I)-Ceva conjugate of X(J) for these (I,J): (69,329), (72,2), (312,346), (314,312), (333,9)

X(8) = X(I)-cross conjugate of X(J) for these (I,J):
(1,280), (9,2), (10,318), (11,522), (55,281), (72,78), (200,346), (210,9), (219,345), (497,7), (521,100)

X(8) = cevapoint of X(I) and X(J) for these (I,J):
(1,40), (2,144), (6,197), (9,200), (10,72), (11,522), (55,219), (175,176)

X(8) = crosspoint of X(I) and X(J) for these (I,J): (75,312), (314,333)

X(8) = crosssum of X(I) and X(J) for these (I,J):
(1,978), (31,604), (57,1423), (667,1015), (1042,1410), (1400,1402)

X(8) = crossdifference of any two points on line X(649)X(854)
X(8) = X(1)-aleph conjugate of X(1050)
X(8) = X(I)-beth conjugate of X(J) for these (I,J): (8,1), (341,341), (643,3), (668,8), (1043,8)


X(9) = MITTENPUNKT

Trilinears       b + c - a : c + a - b : a + b - c
                        = cot(A/2) : cot(B/2) : cot(C/2)

Barycentrics  a(b + c - a) : b(c + a - b) : c(a + b - c)

X(9) is the symmedian point of the excentral triangle.

If you have The Geometer's Sketchpad, you can view Mittenpunkt.

X(9) lies on these lines:
1,6   2,7   3,84   4,10   5,1729   8,346   21,41   31,612   32,987   33,212   34,201   35,90   38,614      39,978   42,941   43,256   46,79   48,101   55,200   56,1696   58,975   100,1005   164,168   165,910   171,1707   173,177   192,239   223,1073   228,1011   241,269   261,645   294,1253   312,314   318,1896   321,1751   342,653   348,738   364,366   374,517   393,1785   440,1211   478,1038   498,920   522,657   604,1420   607,1039   608,1041   609,1333   644,1320   654,1639   750,896   943,1802   986,1722   991,1818   1088,1223   1125,1732   1174,1621   1249,1712   1377,1703   1378,1702   1479,1752   1571,1574   1572,1573   1678,1705   1679,1704   1680,1701   1681,1700

X(9) is the {X(44),X(45)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(9), click More at the top of this page. X(9) is the internal center of similitude of the Bevan circle and Spieker circle; the external center is X(1706).

X(9) = midpoint of X(I) and X(J) for these (I,J): (7,144), (8,390)
X(9) = reflection of X(I) in X(J) for these (I,J): (1,1001), (7,142)
X(9) = isogonal conjugate of X(57)
X(9) = isotomic conjugate of X(85)
X(9) = complement of X(7)
X(9) = anticomplement of X(142)
X(9) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,1), (8,200), (21,55), (63,40), (190,522), (312,78), (318,33), (329, 1490), (333,8)

X(9) = cevapoint of X(I) and X(J) for these (I,J): (1,165), (6,198), (31,205), (37,71), (41,212), (55,220)

X(9) = X(I)-cross conjugate of X(J) for these (I,J):
(6,282), (37,281), (41,33), (55,1), (71,219), (210,8), (212,78), (220,200)

X(9) = crosspoint of X(I) and X(J) for these (I,J): (2,8), (21,333), (63,271), (312,318)
X(9) = crosssum of X(I) and X(J) for these (I,J): (6,56), (19,208), (65,1400), (244,649), (603,604), (1418,1475)
X(9) = crossdifference of any two points on line X(513)X(663)
X(9) = X(I)-Hirst inverse of X(J) for these (I,J): (1, 518), (192,239)

X(9) = X(I)-aleph conjugate of X(J) for these (I,J):
(1,43), (2,9), (9,170), (188,165), (190,1018), (366,1), (507,361), (508,57), (509,978)

X(9) = X(I)-beth conjugate of X(J) for these (I,J):
(9,6), (190,6), (346,346), (644,9), (645,75)


X(10) = SPIEKER CENTER

Trilinears       bc(b + c) : ca(c + a) : ab(a + b)
Barycentrics  b + c : c + a : a + b

The Spieker circle is the incircle of the medial triangle; its center, X(10), is the centroid of the perimeter of ABC.

If you have The Geometer's Sketchpad, you can view Spieker center.

X(10) lies on these lines:
1,2   3,197   4,9   5,517   6,1377   11,121   12,65   20,165   21,35   28,1891   29,1794   31,964   33,406   34,475   36,404   37,594   38,596   39,730   44,752   46,63   55,405   56,474   57,388   58,171   69,969   75,76   81,1224   82,83   86,319   87,979   92,1838   98,101   106,1222   116,120   117,123   119,124   140,214   141,142   150,1282   153,1768   158,318   182,1678   190,671   191,267   201,225   219,965   227,1214   235,1902   255,1771   257,1581   261,1326   274,291   307,1254   321,756   348,1323   391,1743   407,1867   427,1829   429,1824   480,954   485,1686   486,1685   497,1697   514,764   535,1155   537,1086   626,760   631,944   632,1483   750,1150   774,1736   775,801   846,1247   894,1046   908,994   962,1695   1018,1334   1074,1735   1146,1212   1482,1656   1587,1703   1588,1702   1762,1782   1828,1883   1900,1904

X(10) is the {X(1),X(2)}-harmonic conjugate of X(1125). For a list of other harmonic conjugates of X(10), click More at the top of this page. X(10) is the internal center of similitude of the Apollonius and nine-points circles.

X(10) = midpoint of X(I) and X(J) for these (I,J): (1,8), (3,355), (4,40), (65,72), (80,100)
X(10) = reflection of X(I) in X(J) for these (I,J): (1,1125), (551,2), (946,5), (1385,140)
X(10) = isogonal conjugate of X(58)
X(10) = isotomic conjugate of X(86)
X(10) = inverse-in-circumcircle of X(1324)
X(10) = complement of X(1)
X(10) = anticomplement of X(1125)
X(10) = complementary conjugate of X(10)

X(10) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,37), (8,72), (75,321), (80,519), (100,522), (313,306)

X(10) = cevapoint of X(I) and X(J) for these (I,J): (1,191), (6,199), (12,201), (37,210), (42,71), (65,227)
X(10) = X(I)-cross conjugate of X(J) for these (I,J): (37,226), (71,306), (191,502), (201,72)
X(10) = crosspoint of X(I) and X(J) for these (I,J): (2,75), (8,318)
X(10) = crosssum of X(I) and X(J) for these (I,J): (6,31), (56,603)
X(10) = crossdifference of any two points on line X(649)X(834)
X(10) = X(I)-beth conjugate of X(J) for these (I,J): (8,10), (10,65), (100,73), (318,225), (643,35), (668,349)


X(11) = FEUERBACH POINT

Trilinears       1 - cos(B - C) : 1 - cos(C - A) : 1 - cos(A - B)
                        = f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin2(B/2 - C/2)
                        = g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc(b + c - a)(b - c)2

Barycentrics  a(1 - cos(B - C)) : b(1 - cos(C - A)) : c(1 - cos(A - B))
                         = h(a,b,c) : h(b,c,a) : h(c,a,b), where h(A,B,C) = (b + c - a)(b - c)2

X(11) is the point of tangency of the nine-point circle and the incircle. The nine-point circle is the circumcenter of the medial triangle, as well as the orthic triangle. Feuerbach's famous theorem states that the nine-point circle is tangent to the incircle and the three excircles.

If you have The Geometer's Sketchpad, you can view Feuerbach point.

X(11) is the {X(1),X(5)}-harmonic conjugate of X(12). For a list of other harmonic conjugates of X(11), click More at the top of this page.

X(11) lies on these lines:
1,5   2,55   3,499   4,56   7,658   8,1320   10,121   13,202   14,203   28,1852   30,36   33,427   34,235   35,140   57,1360   65,117   68,1069   110,215   113,942   115,1015   118,226   124,1364   182,1848   133,1838   153,388   212,748   214,442   244,867   278,1857   325,350   381,999   403,1870   429,1104   485,1124   486,1335   498,1656   515,1319   516,1155   517,1737   518,908   523,1090   613,1352   650,1566   774,1393   944,1388   962,1788   971,1538   1012,1470   1040,1368   1111,1358   1146,1639   1193,1834   1427,1856   1428,1503   1455,1877   1500,1506   1697,1698

X(11) = midpoint of X(I) and X(J) for these (I,J): (1,80), (4,104), (5,1484), (100,149)
X(11) = reflection of X(I) in X(J) for these (I,J): (1,1387), (119,5), (214,1125), (1145,10), (1317,1), (1537,946)
X(11) = isogonal conjugate of X(59)
X(11) = inverse-in-Furhmann-circle of X(1837)
X(11) = complement of X(100)
X(11) = anticomplement of X(3035)
X(11) = complementary conjugate of X(513)
X(11) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,523), (4,513), (7,514), (8,522), (262,1491)
X(11) = crosspoint of X(I) and X(J) for these (I,J): (7,514), (8,522)
X(11) = crosssum of X(I) and X(J) for these (I,J): (6,692), (55,101), (56,109), (1381,1382), (1397,1415)
X(11) = crossdifference of any two points on line X(101)X(109)
X(11) = X(I)-beth conjugate of X(J) for these (I,J): (11,244), (522,11), (693,11)


X(12) = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11)

Trilinears       1 + cos(B - C) : 1 + cos(C - A) : 1 + cos(A - B)
                        = f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos2(B/2 - C/2)
                        = g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc(b + c)2/(b + c - a)

Barycentrics  a(1 + cos(B - C)) : b(1 + cos(C - A)) : c(1 + cos(A - B))
                         = h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = (b + c)2/(b + c - a)

Let A'B'C' be the touch points of the nine-point circle with the A-, B-, C- excircles, respectively. The lines AA', BB', CC' concur in X(12).

X(12) lies on these lines:
1,5   2,56   3,498   4,55   7,1268   10,65   17,203   18,202   30,35   33,235   34,427   36,140   37,225   38,1393   40,1836   42,1834   54,215   57,1224   63,1454   71,1901   79,484   85,120   108,451   115,1500   116,1362   117,1364   121,1357   123,1359   124,1361   125,1425   141,1469   171,1399   172,230   201,756   208,1360   221,1853   228,407   281,1118   313,349   354,1210   377,1259   381,1479   431,1824   443,1466   474,1470   485,1124   486,1124   499,999   603,750   611,1352   908,960   942,1209   946,1532   968,1904   1015,1506   1038,1368   1091,1109   1125,1319   1213,1400   1452,1892   1594,1870   1697,1699   1861,1887   1877,1883

X(12) is the {X(1),X(5)}-harmonic conjugate of X(11). For a list of other harmonic conjugates of X(12), click More at the top of this page.

If you have The Geometer's Sketchpad, you can view X(12).

X(12) = isogonal conjugate of X(60)
X(12) = isotomic conjugate of X(261)
X(12) = complement of X(2975)
X(12) = X(10)-Ceva conjugate of X(201)
X(12) = crosssum of X(58) and X(1437)
X(12) = X(I)-beth conjugate of X(J) for these (I,J): (10,12), (1089,1089)


X(13) = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)

Trilinears       csc(A + π/3) : csc(B + π/3) : csc(C + π/3)
                        = sec(A - π/6) : sec(B - π/6) : sec(C - π/6)

Barycentrics  f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = a4 - 2(b2 - c2)2 + a2(b2 + c2 + 4*sqrt(3)*Area(ABC))

Construct the equilateral triangle BA'C having base BC and vertex A' on the negative side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA',BB',CC' concur in X(13). If each of the angles A, B, C is < 2*π/3, then X(13) is the only center X for which the angles BXC, CXA, AXB are equal, and X(13) minimizes the sum |AX| + |BX| + |CX|. The antipedal triangle of X(13) is equilateral.

If you have The Geometer's Sketchpad, you can view these sketches:
1st isogonic center
Kiepert Hyperbola, showing X(13) and X(14) on the hyperbola, with midpoint X(115).
Evans Conic, passing through X(13), X(14), X(15), X(16), X(17), X(18), X(3070), X(3071).
X(3054), center of the Evans Conic and 19 other triangle centers.

The Evans conic is introduced in

Evans, Lawrence S., "A Conic Through Six Triangle Centers," Forum Geometricorum 2 (2002) 89-92.

X(13) lies on these lines:
2,16   3,17   4,61   5,18   6,14   11,202   15,30   76,299   80,1251   98,1080   99,303   148,617   203,1478   226,1081   262,383   275,472   298,532   484,1277   531,671   533,621   634,635

X(13) is the {X(6),X(381)}-harmonic conjugate of X(14). For a list of other harmonic conjugates of X(13), click More at the top of this page.

X(13) = reflection of X(I) in X(J) for these (I,J): (14,115), (15,396), (99,619), (298,623), (616,618)
X(13) = isogonal conjugate of X(15)
X(13) = isotomic conjugate of X(298)
X(13) = inverse-in-orthocentroidal-circle of X(14)
X(13) = complement of X(616)
X(13) = anticomplement of X(618)
X(13) = cevapoint of X(15) and X(62)
X(13) = X(I)-cross conjugate of X(J) for these (I,J): (15,18), (30,14), (396,2)


X(14) = 2nd ISOGONIC CENTER

Trilinears       csc(A - π/3) : csc(B - π/3) : csc(C - π/3)
                        = sec(A + π/6) : sec(B + π/6) : sec(C + π/6)

Barycentrics  f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = a4 - 2(b2 - c2)2 + a2(b2 + c2 - 4*sqrt(3)*Area(ABC))

Construct the equilateral triangle BA'C having base BC and vertex A' on the positive side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA',BB',CC' concur in X(14). The antipedal triangle of X(14) is equilateral.

If you have The Geometer's Sketchpad, you can view 2nd isogonic center

X(14) lies on these lines:
2,15   3,18   4,62   5,17   6,13   11,203   16,30   76,298   98,383   99,302   148,616   202,1478   226,554   262,1080   275,473   299,533   397,546   484,1276   530,671   532,622   633,636

X(14) is the {X(6),X(381)}-harmonic conjugate of X(13). For a list of other harmonic conjugates of X(14), click More at the top of this page.

X(14) = reflection of X(I) in X(J) for these (I,J): (13,115), (16,395), (99,618), (299,624), (617,619)
X(14) = isogonal conjugate of X(16)
X(14) = isotomic conjugate of X(299)
X(14) = inverse-in-orthocentroidal-circle of X(13)
X(14) = complement of X(617)
X(14) = anticomplement of X(619)
X(14) = cevapoint of X(16) and X(61)
X(14) = X(I)-cross conjugate of X(J) for these (I,J): (16,17), (30,13), (395,2)


X(15) = 1st ISODYNAMIC POINT

Trilinears       sin(A + π/3) : sin(B + π/3) : sin(C + π/3)
                        = cos(A - π/6) : cos(B - π/6) : cos(C - π/6)

Barycentrics  a sin(A + π/3) : b sin(B + π/3) : c sin(C + π/3)

Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through one vertex and both isodynamic points. (This configuration, based on the cevians of X(1), generalizes to arbitrary cevians; see TCCT, p. 98, problem 8.)

The pedal triangle of X(15) is equilateral.

If you have The Geometer's Sketchpad, you can view 1st isodynamic point and X(15)&X(16), with Brocard axis and Lemoine axis.

X(15) lies on these lines:
1,1251   2,14   3,6   4,17   13,30   18,140   35,1250   36,202   55,203   298,533   303,316   395,549   397,550   532,616   628,636

X(15) is the {X(3),X(6)}-harmonic conjugate of X(16). For a list of other harmonic conjugates of X(15), click More at the top of this page.

X(15) = reflection of X(I) in X(J) for these (I,J): (13,396), (16,187), (298,618), (316,624), (621,623)
X(15) = isogonal conjugate of X(13)
X(15) = isotomic conjugate of X(300)
X(15) = inverse-in-circumcircle of X(16)
X(15) = inverse-in-Brocard-circle of X(16)
X(15) = complement of X(621)
X(15) = anticomplement of X(623)
X(15) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,202), (13,62), (74,16)
X(15) = crosspoint of X(I) and X(J) for these (I,J): (13,18), (298,470)
X(15) = crosssum of X(I) and X(J) for these (I,J): (15,62), (532,619)
X(15) = crossdifference of any two points on line X(395)X(523)
X(15) = X(6)-Hirst inverse of X(16)


X(16) = 2nd ISODYNAMIC POINT

Trilinears       sin(A - π/3) : sin(B - π/3) : sin(C - π/3)
                        = cos(A + π/6) : cos(B + π/6) : cos(C + π/6)

Barycentrics  a sin(A - π/3) : b sin(B - π/3) : c sin(C - π/3)

Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through a vertex and both isodynamic points. The pedal triangle of X(16) is equilateral.

If you have The Geometer's Sketchpad, you can view 2nd isodynamic point.

X(16) lies on these lines:
1,1250   2,13   3,6   4,18   14,30   17,140   36,203   55,202   299,532   302,316   358,1135   396,549   398,550   533,617   627,635

X(16) is the {X(3),X(6)}-harmonic conjugate of X(15). For a list of other harmonic conjugates of X(16), click More at the top of this page.

X(16) = reflection of X(I) in X(J) for these (I,J): (14,395), (15,187), (299,619), (316,623), (622,624)
X(16) = isogonal conjugate of X(14)
X(16) = isotomic conjugate of X(301)
X(16) = inverse-in-circumcircle of X(15)
X(16) = inverse-in-Brocard-circle of X(15)
X(16) = complement of X(622)
X(16) = anticomplement of X(624)
X(16) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,203), (14,61), (74,15)
X(16) = crosspoint of X(I) and X(J) for these (I,J): (14,17), (299,471)
X(16) = crosssum of X(I) and X(J) for these (I,J): (16,61), (533,618)
X(16) = crossdifference of any two points on line X(396)X(523)
X(16) = X(6)-Hirst inverse of X(15)


X(17) = 1st NAPOLEON POINT

Trilinears       csc(A + π/6) : csc(B + π/6) : csc(C + π/6)
                        = sec(A - π/3) : sec(B - π/3) : sec(C - π/3)

Barycentrics  a csc(A + π/6) : b csc(B + π/6) : c csc(C + π/6)

Let X,Y,Z be the centers of the equilateral triangles in the construction of X(13). The lines AX, BY, CZ concur in X(17).

John Rigby, "Napoleon revisited," Journal of Geometry, 33 (1988) 126-146.

If you have The Geometer's Sketchpad, you can view 1st Napoleon point.

X(17) lies on these lines:
2,62   3,13   4,15   5,14   6,18   12,203   16,140   76,303   83,624   202,499   275,471   299,635   623,633

X(17) is the {X(231),X(1209)}-harmonic conjugate of X(18). For a list of other harmonic conjugates of X(17), click More at the top of this page.

X(17) = reflection of X(627) in X(629)
X(17) = isogonal conjugate of X(61)
X(17) = isotomic conjugate of X(302)
X(17) = complement of X(627)
X(17) = anticomplement of X(629)
X(17) = X(I)-cross conjugate of X(J) for these (I,J): (16,14), (140,18), (397,4)


X(18) = 2nd NAPOLEON POINT

Trilinears       csc(A - π/6) : csc(B - π/6) : csc(C - π/6)
                        = sec(A + π/3) : sec(B + π/3) : sec(C + π/3)

Barycentrics  a csc(A - π/6) : b csc(B - π/6) : c csc(C - π/6)

Let X,Y,Z be the centers of the equilateral triangles in the construction of X(14). The lines AX, BY, CZ concur in X(18).

If you have The Geometer's Sketchpad, you can view 2nd Napoleon point.

X(18) lies on these lines:
2,61   3,14   4,16   5,13   6,17   12,202   15,140   76,302   83,623   203,499   275,470   298,636   624,634

X(18) is the {X(231),X(1209)}-harmonic conjugate of X(17). For a list of other harmonic conjugates of X(18), click More at the top of this page.

X(18) = reflection of X(628) in X(630)
X(18) = isogonal conjugate of X(62)
X(18) = isotomic conjugate of X(303)
X(18) = complement of X(628)
X(18) = anticomplement of X(630)
X(18) = X(I)-cross conjugate of X(J) for these (I,J): (15,13), (140,17), (398,4)


X(19) = CLAWSON POINT

Trilinears       tan A : tan B : tan C
                        = f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin 2B + sin 2C - sin 2A
                        = g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/(b2 + c2 - a2)

Barycentrics  a tan A : b tan B : c tan C

X(19) is the homothetic center of the orthic and extangents triangles.

If you have The Geometer's Sketchpad, you can view Clawson point.

Further information is available from
Paul Yiu's Website.

Although John Clawson studied this point in 1925, it was studied earlier by Lemoine:

Emile Lemoine, "Quelques questions se rapportant à l'étude des antiparallèles des côtes d'un triangle", Bulletin de la S. M. F., tome 14 (1886), p. 107-128, specifically, on page 114. This article is available online at Numdam.

X(19) lies on these lines:
1,28   2,534   3,1871   4,9   6,34   8,1891   25,33   27,63   31,204   41,1825   44,1828   45,1900   46,579   47,921   53,1846   56,207   57,196   64,1903   81,969   91,920   101,913   102,282   112,759   158,1712   162,897   163,563   208,225   219,517   220,1902   226,1763   232,444   273,653   294,1041   318,1840   379,1441   407,1865   429,1213   560,1910   604,909   672,1851   960,965   1158,1715   1212,1593   1405,1866   1449,1870   1581,1740   1598,1872   1633,1721   1707,1719   1708,1713   1743,1783   1836,1901   1837,1852

X(19) is the {X(607),X(608)}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(19), click More at the top of this page.

X(19) = isogonal conjugate of X(63)
X(19) = isotomic conjugate of X(304)

X(19) = X(I)-Ceva conjugate of X(J) for these (I,J):
(1,204), (4,33), (27,4), (28,25), (57,208), (92,1), (196,207), (278,34)

X(19) = X(I)-cross conjugate of X(J) for these (I,J): (25,34), (31,1)
X(19) = crosspoint of X(I) and X(J) for these (I,J): (4,278), (27,28), (57,84), (92,158)
X(19) = crosssum of X(I) and X(J) for these (I,J): (1,610), (3,219), (9,40), (48,255), (71,72)
X(19) = crossdifference of any two points on line X(521)X(656)
X(19) = X(I)-Hirst inverse of X(J) for these (I,J): (1,240), (4,242)
X(19) = X(I)-aleph conjugate of X(J) for these (I,J): (2,610), (92,19), (508,223), (648,163)
X(19) = X(I)-beth conjugate of X(J) for these (I,J): (9,198), (19,608), (112,604), (281,281), (648,273), (653,19)



leftri Centers 20- 30, rightri
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.

underbar

X(20) = DE LONGCHAMPS POINT

Trilinears       cos A - cos B cos C : cos B - cos C cos A : cos C - cos A cos B
                        = sec A - sec B sec C : sec B - sec C sec A : sec C - sec A sec B

Barycentrics  tan B + tan C - tan A : tan C + tan A - tan B : tan A + tan B - tan C
                        = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [-3a4 + 2a2(b2 + c2) + (b2 - c2)2]

X(2) is the reflection of X(4) in X(3); also, the orthocenter of the anticomplementary triangle.

If you have The Geometer's Sketchpad, you can view De Longchamps point.

X(20) lies on these lines:
1,7   2,3   8,40   10,165   33,1038   34,1040   35,1478   36,1479   55,388   56,497   57,938   58,387   64,69   68,74   72,144   78,329   97,1217   98,148   99,147   100,153   101,152   103,150   104,149   109,151   110,146   145,517   155,323   185,193   190,1265   243,1118   254,1300   346,1766   371,1587   372,1588   391,573   393,577   394,1032   485,1131   486,1132   487,638   488,637   616,633   617,635   621,627   622,628   936,1750   999,1058   1062,1870   1074,1838   1076,1785   1125,1699   1147,1614   1155,1788   1204,1899   1440,1804   1610,1633

X(20) is the {X(3),X(4)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(20), click More at the top of this page.

X(20) = reflection of X(I) in X(J) for these (I,J): (2,376), (3,550), (4,3), (5,548), (8,40), (69,1350), (145,944), (146,110), (147,99), (148,98), (149,104), (150,103), (151,109), (152,101), (153,100), (382,5), (962,1)

X(20) = isogonal conjugate of X(64)
X(20) = isotomic conjugate of X(253)
X(20) = cyclocevian conjugate of X(1032)
X(20) = inverse-in-circumcircle of X(2071)
X(20) = inverse-in-orthocentroidal-circle of X(3091)
X(20) = complement of X(3146)
X(20) = anticomplement of X(4)
X(20) = anticomplementary conjugate of X(4)
X(20) = X(I)-Ceva conjugate of X(J) for these (I,J): (69,2), (489,487), (490,488)
X(20) = crosssum of X(1) and X(1044)
X(20) = crossdifference of any two points on line X(647)X(657)
X(20) = X(I)-aleph conjugate of X(J) for these (I,J): (8,191), (9,1045), (188,1046), (333,2), (1043,20)
X(20) = X(I)-beth conjugate of X(J) for these (I,J): (664,20), (1043,280)


X(21) = SCHIFFLER POINT

Trilinears       1/(cos B + cos C) : 1/(cos C + cos A) : 1/(cos A + cos B)
                        = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b + c)

Barycentrics  a/(cos B + cos C) : b/(cos C + cos A) : c/(cos A + cos B)

Write I for the incenter; the Euler lines of the four triangles IBC, ICA, IAB, and ABC concur in X(21).

If you have The Geometer's Sketchpad, you can view Schiffler point.

Lev Emelyanov and Tatiana Emelyanova, A note on the Schiffler point, Forum Geometricorum 3 (2003) pages 113-116.

The name of this point honors Kurt Schiffler.

X(21) lies on these lines:
1,31   2,3   6,941   7,56   8,55   9,41   10,35   32,981   36,79   37,172   51,970   60,960   72,943   75,272   77,1394   84,285   90,224   99,105   104,110   107,1295   144,954   145,956   238,256   243,1896   261,314   268,280   270,1172   286,1441   294,1212   332,1036   385,1655   386,1724   517,1389   572,1765   600,1698   612,989   614,988   643,1320   644,1334   662,1156   741,932   748,978   884,885   915,925   961,1402   976,983   1030,1213   1038,1041   1039,1040   1060,1063   1061,1062   1214,1396   1254,1758   1319,1408   1412,1420

X(21) is the {X(2),X(3)}-harmonic conjugate of X(404). For a list of other harmonic conjugates of X(21), click More at the top of this page.

X(21) = midpoint of X(1) and X(191)
X(21) = isogonal conjugate of X(65)
X(21) = isotomic conjugate of X(1441)
X(21) = inverse-in-circumcircle of X(1325)
X(21) = anticomplement of X(442)
X(21) = X(I)-Ceva conjugate of X(J) for these (I,J): (86,81), (261,333)
X(21) = cevapoint of X(I) and X(J) for these (I,J): (1,3), (9,55)

X(21) = X(I)-cross conjugate of X(J) for these (I,J):
(1,29), (3,283), (9,333), (55,284), (58,285), (284,81), (522,100)

X(21) = crosspoint of X(86) and X(333)
X(21) = crosssum of X(I) and X(J) for these (I,J): (1,1046), (42,1400), (1254,1425), (1402,1409)
X(21) = crossdifference of any two points on line X(647)X(661)
X(21) = X(I)-Hirst inverse of X(J) for these (I,J): (2,448), (3,416), (4,425)
X(21) = X(I)-beth conjugate of X(J) for these (I,J): (21,58), (99,21), (643,21), (1043,1043), (1098,21)


X(22) = EXETER POINT

Trilinears       a(b4 + c4 - a4) : b(c4 + a4 - b4) : c(a4 + b4 - c4)
Barycentrics  a2(b4 + c4 - a4) : b2(c4 + a4 - b4) : c2(a4 + b4 - c4)
Barycentrics  sin 2A - tan ω : sin 2B - tan ω : sin 2C - tan ω     (M. Iliev, 5/13/07)

X(22) is the perspector of the circummedial triangle and the tangential triangle; also X(22) = X(55)-of-the-tangential-triangle if ABC is acute. See the note just before X(1601) for a generalization.

If you have The Geometer's Sketchpad, you can view Exeter point.

X(22) lies on these lines:
2,3   6,251   32,1194   35,612   36,614   51,182   56,977   69,159   76,1799   98,925   99,305   100,197   110,154   155,1614   157,183   160,325   161,343   184,511   187,1196   232,577   264,1629   347,1617   675,1305   991,1790   1184,1627   1294,1302   1486,1621   1602,1626

X(22) is the {X(3),X(25)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(22), click More at the top of this page.

X(22) = reflection of X(378) in X(3)
X(22) = isogonal conjugate of X(66)
X(22) = inverse-in-circumcircle of X(858)
X(22) = anticomplement of X(427)
X(22) = X(76)-Ceva conjugate of X(6)
X(22) = cevapoint of X(3) and X(159)
X(22) = crosspoint of X(99) and X(250)
X(22) = crosssum of X(125) and X(512)
X(22) = crossdifference of any two points on line X(647)X(826)
X(22) = X(I)-beth conjugate of X(J) for these (I,J): (643,345), (833,22)


X(23) = FAR-OUT POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b4 + c4 - a4 - b2c2]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics  2 sin 2A - 3 tan ω : 2 sin 2B - 3 tan ω : 2 sin 2C - 3 tan ω     (M. Iliev, 5/13/07)

X(23) is the inverse-in-circumcircle of X(2).

If you have The Geometer's Sketchpad, you can view Far-out point.

X(23) lies on these lines:
2,3   6,353   51,575   52,1614   94,98   105,1290   110,323   111,187   143,1199   159,193   184,576   232,250   251,1194   324,1629   385,523   477,1302   895,1177   1196,1627   1297,1804

X(23) is the {X(22),X(25)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(23), click More at the top of this page.

X(23) = reflection of X(I) in X(J) for these (I,J): (110,1495), (323,110), (691,187), (858,468)
X(23) = isogonal conjugate of X(67)
X(23) = inverse-in-circumcircle of X(2)
X(23) = anticomplement of X(858)
X(23) = crosspoint of X(111) and X(251)
X(23) = crosssum of X(I) and X(J) for these (I,J): (125,690), (141,524)
X(23) = crossdifference of any two points on line X(39)X(647)


X(24) = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE

Trilinears       sec A cos 2A : sec B cos 2B : sec C cos 2C
                        = sec A - 2 cos A : sec B - 2 cos B : sec C - 2 cos C

Barycentrics   tan A cos 2A : tan B cos 2B : tan C cos 2C
                        = tan A - sin 2A : tan A - sin 2B : tan C - sin 2C

Constructed as indicated by the name; also X(24) = X(56)-of-the-tangential triangle if ABC is acute.

If you have The Geometer's Sketchpad, you can view X(24).

X(24) lies on these lines:
1,1061   2,3   6,54   32,232   33,35   34,36   49,568   51,578   52,1147   56,1870   64,74   96,847   98,1289   107,1093   108,915   110,155   154,1181   182,1843   183,1235   184,389   185,1495   242,1602   254,393   264,1078   511,1092   573,1474   602,1395   944,1610   1063,1775   1112,1511   1192,1511   1324,1603   1385,1829

X(24) is the {X(3),X(4)}-harmonic conjugate of X(378). For a list of other harmonic conjugates of X(24), click More at the top of this page.

X(24) = reflection of X(4) in X(235)
X(24) = isogonal conjugate of X(68)
X(24) = inverse-in-circumcircle of X(403)
X(24) = X(249)-Ceva conjugate of X(112)
X(24) = X(52)-cross conjugate of X(4)
X(24) = crosspoint of X(107) and X(250)
X(24) = crosssum of X(I) and X(J) for these (I,J): (6,161), (125,520), (637,638)
X(24) = X(4)-Hirst inverse of X(421)


X(25) = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES

Trilinears       sin A tan A : sin B tan B : sin C tan C = cos A - sec A : cos B - sec B : cos C - sec C
                        = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(b2 + c2 - a2)

Barycentrics  sin 2A - 2 tan A : sin 2B - 2 tan B : sin 2C - 2 tan C
                        = g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/(b2 + c2 - a2)

Constructed as indicated by the name; also X(25) is the pole of the orthic axis (the line having trilinear coefficients cos A : cos B : cos C) with respect to the circumcircle. Also, X(25) is X(57)-of-the-tangential triangle.

If you have The Geometer's Sketchpad, you can view X(25).

X(25) lies on these lines:
1,1036   2,3   6,51   19,33   31,608   32,1184   34,56   35,1900   36,1878   40,1902   41,42   52,155   53,157   57,1473   58,967   64,1192   65,1452   76,1241   92,242   98,107   100,1862   105,108   110,1112   111,112   114,135   125,1853   132,136   143,156   183,264   185,1498   221,1425   225,1842   226,1892   262,275   273,1447   286,1218   317,325   339,1289   343,1352   371,493   372,494   389,1181   393,1033   394,511   669,878   692,913   694,1613   842,1304   847,1179   941,1172   958,1891   999,1870   1001,1848   1073,1297   1096,1402   1235,1239   1300,1302   1324,1785   1376,1861   1470,1877   1503,1619   1604,1863   1631,1826   1726,1736   1730,1754

X(25) is the {X(5),X(26)}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(25), click More at the top of this page.

X(25) = reflection of X(I) in X(J) for these (I,J): (4,1596), (1370,1368)
X(25) = isogonal conjugate of X(69)
X(25) = isotomic conjugate of X(305)
X(25) = inverse-in-circumcircle of X(468)
X(25) = inverse-in-orthocentroidal-circle of X(427)
X(25) = complement of X(1370)
X(25) = anticomplement of X(1368)
X(25) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,6), (28,19), (250,112)
X(25) = X(32)-cross conjugate of X(6)
X(25) = crosspoint of X(I) and X(J) for these (I,J): (4,393), (6,64), (19,34), (112,250)
X(25) = crosssum of X(I) and X(J) for these (I,J): (2,20), (3,394), (6,159), (8,329), (63,78), (72,306), (125,525)
X(25) = crossdifference of any two points on line X(441)X(525)
X(25) = X(I)-Hirst inverse of X(J) for these (I,J): (4,419), (6,232)
X(25) = X(I)-beth conjugate of X(J) for these (I,J): (33,33), (108,25), (162,278)


X(26) = CIRCUMCENTER OF THE TANGENTIAL TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b2cos 2B + c2cos 2C - a2cos 2A]
                         Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (J2 - 3) cos A + 4 cos B cos C, where J is as at X(1113)

Barycentrics  h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = a2(b2cos 2B + c2cos 2C - a2cos 2A)

If you have The Geometer's Sketchpad, you can view X(26).

Theorems involving X(26), published in 1889 by A. Gob, are discussed in
Roger A. Johnson, Advanced Euclidean Geometry, Dover, 1960, 259-260.

X(26) lies on these lines: 2,3   6,143   52,184   68,161   98,1286   154,155   206,511   1605,1607   1606,1608

X(26) is the {X(154),X(155)}-harmonic conjugate of X(156). For a list of other harmonic conjugates of X(26), click More at the top of this page.

X(26) = reflection of X(155) in X(156)
X(26) = isogonal conjugate of X(70)
X(26) = inverse-in-circumcircle of X(2072)
X(26) = crosssum of X(125) and X(924)


X(27) = CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER

Trilinears       (sec A)/(b + c) : (sec B)/(c + a) : (sec C)/(a + b)
Barycentrics  (tan A)/(b + c) : (tan B)/(c + a) : (tan C)/(a + b)

If you have The Geometer's Sketchpad, you can view X(27).

X(27) lies on these lines:
2,3   6,1246   7,81   19,63   57,273   58,270   84,1896   86,1474   103,107   110,917   226,284   239,1829   243,1859   295,335   306,1043   393,967   579,1751   648,903   662,913   1014,1440   1088,1434   1268,1796   1719,1733   1730,1746   1770,1780

X(27) is the {X(2),X(4)}-harmonic conjugate of X(469). For a list of other harmonic conjugates of X(27), click More at the top of this page.

X(27) = isogonal conjugate of X(71)
X(27) = isotomic conjugate of X(306)
X(27) = inverse-in-orthocentroidal-circle of X(469)
X(27) = complement of X(3151)
X(27) = anticomplement of X(440)
X(27) = X(286)-Ceva conjugate of X(29)
X(27) = cevapoint of X(I) and X(J) for these (I,J): (4,19), (57,278)
X(27) = X(I)-cross conjugate of X(J) for these (I,J): (4,286), (19,28), (57,81), (58,86)
X(27) = crossdifference of any two points on line X(647)X(810)
X(27) = X(I)-Hirst inverse of X(J) for these (I,J): (2,447), (4,423)
X(27) = X(I)-beth conjugate of X(J) for these (I,J): (648,27), (923,27)


X(28) = CEVAPOINT OF X(19) AND X(25)

Trilinears       (tan A)/(b + c) : (tan B)/(c + a) : (tan C)/(a + b)
Barycentrics  (sin A tan A)/(b + c) : (sin B tan B)/(c + a) : (sin C tan C)/(a + b)

If you have The Geometer's Sketchpad, you can view X(28).

X(28) lies on these lines:
1,19   2,3   10,1891   11,1852   33,975   34,57   35,1869   36,1838   46,1780   54,1243   56,278   60,81   65,1175   72,1257   88,162   104,107   105,112   108,225   110,915   142,1890   228,943   242,261   272,273   279,1014   281,958   291,1783   501,1831   579,1724   580,1730   607,1002   608,959   614,1472   956,1219   957,1191   961,1169   1104,1333   1125,1848   1155,1888   1170,1876   1178,1432   1224,1826   1255,1824   1295,1301   1385,1871   1412,1422   1633,1770   1710,1725

X(28) is the {X(27),X(29)}-harmonic conjugate of X(4). For a list of other harmonic conjugates of X(28), click More at the top of this page.

X(28) = isogonal conjugate of X(72)
X(28) = X(I)-Ceva conjugate of X(J) for these (I,J): (270,58), (286,81)
X(28) = cevapoint of X(I) and X(J) for these (I,J): (19,25), (34,56)
X(28) = X(I)-cross conjugate of X(J) for these (I,J): (19,27), (58,58)
X(28) = crossdifference of any two points on line X(647)X(656)
X(28) = X(4)-Hirst inverse of X(422)
X(28) = X(I)-beth conjugate of X(J) for these (I,J): (29,29), (107,28), (162,28), (270,28)


X(29) = CEVAPOINT OF INCENTER AND ORTHOCENTER

Trilinears       (sec A)/(cos B + cos C) : (sec B)/(cos C + cos A) : (sec C)/(cos A + cos B)
Barycentrics  (tan A)/(cos B + cos C) : (tan B)/(cos C + cos A) : (tan C)/(cos A + cos B)

If you have The Geometer's Sketchpad, you can view X(29).

X(29) lies on these lines:
1,92   2,3   8,219   10,1794   33,78   34,77   58,162   65,296   81,189   102,107   112,1311   226,951   242,257   270,283   284,950   314,1039   388,1037   392,1871   497,1036   515,947   648,1121   662,1800   758,1844   894,1868   960,1859   1056,1059   1057,1058   1125,1838   1220,1474   1737,1780   1807,1897   1842,1848

X(29) is the {X(3),X(4)}-harmonic conjugate of X(412). For a list of other harmonic conjugates of X(29), click More at the top of this page.

X(29) = isogonal conjugate of X(73)
X(29) = isotomic conjugate of X(307)
X(29) = complement of X(3153)
X(29) = X(286)-Ceva conjugate of X(27)
X(29) = cevapoint of X(I) and X(J) for these (I,J): (1,4), (33,281)
X(29) = X(I)-cross conjugate of X(J) for these (I,J): (1,21), (284,333), (497,314)
X(29) = crosssum of X(I) and X(J) for these (I,J): (1,1047), (228,1409)
X(29) = crossdifference of any two points on line X(647)X(822)
X(29) = X(4)-Hirst inverse of X(415)
X(29) = X(I)-beth conjugate of X(J) for these (I,J): (29,28), (811,29)


X(30) = EULER INFINITY POINT

Trilinears       cos A - 2 cos B cos C : cos B - 2 cos C cos A : cos C - 2 cos A cos B
                        = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[2a4 - (b2 - c2)2 - a2(b2 + c2)]

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2a4 - (b2 - c2)2 - a2(b2 + c2)

X(30) is the point of intersection of the Euler line and the line at infinity. Thus, each of the 41 lines listed below is parallel to the Euler line.

If you have The Geometer's Sketchpad, you can view Euler Infinity Point.

X(30) lies on these lines:
1,79   2,3   11,36   12,35   13,15   14,16   33,1060   34,1062   40,191   46,1837   49,1614   52,185   53,577   55,495   56,496   58,1834   61,397   62,398   64,68   65,1770   74,265   80,484   98,671   99,316   104,1290   110,477   113,1495   115,187   143,389   146,323   148,385   155,1498   156,1147   182,597   262,598   284,1901   298,616   299,617   340,1494   390,1056   485,1151   486,1152   489,638   490,637   497,999   511,512   551,946   553,942   582,1724   599,1350   618,623   619,624   620,625   841,1302   935,1297   944,962   1043,1330   1141,1157   1155,1737   1294,1304   1319,1387   1351,1353   1465,1877   1838,1852

X(30) = orthopoint of X(523)
X(30) = isogonal conjugate of X(74)
X(30) = isotomic conjugate of X(1494)
X(30) = anticomplementary conjugate of X(146)
X(30) = complementary conjugate of X(113)
X(30) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,113), (265,5), (476,523)
X(30) = cevapoint of X(3) and X(399)
X(30) = crosspoint of X(I) and X(J) for these (I,J): (13,14), (94,264)
X(30) = crosssum of X(I) and X(J) for these (I,J): (15,16), (50,184)
X(30) = crossdifference of any two points on line X(6)X(647)


X(31) = 2nd POWER POINT

Trilinears       a2 : b2 : c2
                        = 1 - cos 2A : 1 - cos 2B : 1 - cos 2C

Barycentrics  a3 : b3 : c3

If you have The Geometer's Sketchpad, you can view X(31) (1), X(31) (2), X(31) (3).

X(31) lies on these lines:
1,21   2,171   3,601   6,42   8,987   9,612   10,964   19,204   25,608   32,41   35,386   36,995   40,580   43,100   44,210   48,560   51,181   56,154   57,105   65,1104   72,976   75,82   76,734   91,1087   92,162   99,715   101,609   110,593   163,923   184,604   237,904   404,978   561,722   649,884   669,875   701,789   743,825   745,827   759,994   775,1097   937,1103   940,1001   999,1149   1139,1140

X(31) is the {X(1),X(63)}-harmonic conjugate of X(38). For a list of other harmonic conjugates of X(21), click More at the top of this page.

X(31) = isogonal conjugate of X(75)
X(31) = isotomic conjugate of X(561)
X(31) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,48), (6,41), (9,205), (58,6), (82,1)
X(31) = X(213)-cross conjugate of X(6)
X(31) = crosspoint of X(I) and X(J) for these (I,J): (1,19), (6,56)

X(31) = crosssum of X(I) and X(J) for these (I,J): (1,63), (2,8), (7,347), (10,321), (239,1281), (244,514), (307,1441), (523,1086), (693,1111)

X(31) = crossdifference of any two points on line X(514)X(661)
X(31) = X(1403)-Hirst inverse of X(1428)
X(31) = X(I)-aleph conjugate of X(J) for these (I,J): (82,31), (83,75)
X(31) = X(I)-beth conjugate of X(J) for these (I,J): (21,993), (55,55), (109,31), (110,57), (643,31), (692,31)


X(32) = 3rd POWER POINT

Trilinears       a3 : b3 : c3
                        = sin(A - ω) : sin(B - ω) : sin(C - ω)
                        = sin A + sin(A - 2ω) : sin B + sin(B - 2ω) : sin C + sin(C - 2ω)
                        = cos A - cos(A - 2ω) : cos B - cos(B - 2ω) : cos C - cos(C - 2ω) (cf., X(39))

Barycentrics  a4 : b4 : c4

If you have The Geometer's Sketchpad, you can view X(32).

X(32) lies on these lines:
1,172   2,83   3,6   4,98   5,230   9,987   21,981   24,232   31,41   56,1015   75,746   76,384   81,980   99,194   100,713   101,595   110,729   163,849   184,211   218,906   512,878   538,1003   561,724   590,640   604,1106   615,639   731,825   733,827   910,1104   993,1107

X(32) is the {X(3),X(6)}-harmonic conjugate of X(39). For a list of other harmonic conjugates of X(21), click More at the top of this page.

X(32) = midpoint of X(371) and X(372)
X(32) = reflection of X(315) in X(626)
X(32) = isogonal conjugate of X(76)
X(32) = isotomic conjugate of X(1502)
X(32) = inverse-in-circumcircle of X(1691)
X(32) = inverse-in-Brocard-circle of X(39)
X(32) = inverse-in-1st-Lemoine-circle of X(1692)
X(32) = complement of X(315)
X(32) = anticomplement of X(626)
X(32) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,206), (6,184), (112,512), (251,6)
X(32) = crosspoint of X(I) and X(J) for these (I,J): (2,66), (6,25)

X(32) = crosssum of X(I) and X(J) for these (I,J): (2,69), (6,22), (75,312), (115,826), (311,343), (313,321), (338,850), (339,525), (349,1231), (693,1086), (1229,1233), (1230,1269)

X(32) = crossdifference of any two points on line X(325)X(523)
X(32) = X(184)-Hirst inverse of X(237)
X(32) = X(I)-beth conjugate of X(J) for these (I,J): (41,41), (163,56), (919,32)
X(32) = external center of similitude of circumcircle and Moses circle


X(33) = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES

Trilinears       1 + sec A : 1 + sec B : 1 + sec C = tan A cot(A/2) : tan B cot(B/2) : tan C cot(C/2)
                        = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b2 + c2 - a2)
                        = g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec A cos2(A/2)

Barycentrics  sin A + tan A : sin B + tan B : sin C + tan C
                        = h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = tan A cos2(A/2)

If you have The Geometer's Sketchpad, you can view X(33).

X(33) lies on these lines:
1,4   2,1040   5,1062   6,204   7,1041   8,1039   9,212   10,406   11,427   12,235   19,25   20,1038   24,35   28,975   29,78   30,1060   36,378   40,201   42,393   47,90   56,963   57,103   63,1013   64,65   79,1063   80,1061   84,603   112,609   200,281   210,220   222,971   264,350

X(33) is the {X(1),X(4)}-harmonic conjugate of X(34). For a list of other harmonic conjugates, click More at the top of this page.

X(33) = isogonal conjugate of X(77)
X(33) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,19), (29,281), (318,9)
X(33) = X(I)-cross conjugate of X(J) for these (I,J): (41,9), (42,55)
X(33) = crosspoint of X(I) and X(J) for these (I,J): (1,282), (4,281)
X(33) = crosssum of X(I) and X(J) for these (I,J): (1,223), (3,222), (57,1394), (73,1214)
X(33) = crossdifference of any two points on line X(652)X(905)
X(33) = X(33)-beth conjugate of X(25)


X(34) = X(4)-BETH CONJUGATE OF X(4)

Trilinears       1 - sec A : 1 - sec B : 1 - sec C
                        = tan A tan(A/2) : tan B tan(B/2) : tan C tan(C/2)
                        = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(b + c - a)(b2 + c2 - a2)]
                        = g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec A sin2(A/2)

Barycentrics  sin A - tan A : sin B - tan B : sin C - tan C
                        = h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = tan A sin2(A/2)

X(34) is the center of perspective of the orthic triangle and the reflection in the incenter of the intangents triangle.

If you have The Geometer's Sketchpad, you can view X(34) (1) and X(34) (2).

X(34) lies on these lines:
1,4   2,1038   5,1060   6,19   7,1039   8,1041   9,201   10,475   11,235   12,427   20,1040   24,36   25,56   28,57   29,77   30,1062   35,378   40,212   46,47   55,227   79,1061   80,1063   87,242   106,108   196,937   207,1042   222,942   244,1106   331,870   347,452   860,997

X(34) is the {X(1),X(4)}-harmonic conjugate of X(33). For a list of other harmonic conjugates of X(34), click More at the top of this page.

X(34) = isogonal conjugate of X(78)
X(34) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,207), (4,208), (28,56), (273,57), (278,19)
X(34) = X(25)-cross conjugate of X(19)
X(34) = crosssum of X(219) and X(1260)
X(34) = X(56)-Hirst inverse of X(1430)
X(34) = X(I)-beth conjugate of X(J) for these (I,J):
(1,221), (4,4), (28,34), (29,1), (107,158), (108,34), (110,47), (162,34), (811,34)

X(34) = crossdifference of any two points on line X(521)X(652)


X(35) = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36)

Trilinears       1 + 2 cos A : 1 + 2 cos B : 1 + 2 cos C
                        = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2 + bc)

Barycentrics  sin A + sin 2A : sin B + sin 2B : sin C + sin 2C
                        = g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 + c2 - a2 + bc)

Let A' be the inverse-in-circumcircle of the A-excenter, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(35).

X(35) lies on these lines:
1,3   4,498   8,993   9,90   10,21   11,140   12,30   22,612   24,33   31,386   34,378   37,267   42,58   43,1011   47,212   71,284   72,191   73,74   79,226   172,187   228,846   255,991   376,388   404,1125   411,516   474,1001   495,550   496,549   497,499   500,1154   595,902   950,1006   968,975   1124,1152

If you have The Geometer's Sketchpad, you can view X(35).

X(35) is the {X(1),X(3)}-harmonic conjugate of X(36). For a list of other harmonic conjugates of X(35), click More at the top of this page.

X(35) = isogonal conjugate of X(79)
X(35) = inverse-in-circumcircle of X(484)
X(35) = X(500)-cross conjugate of X(1)
X(35) = crosssum of X(481) and X(482)
X(35) = X(943)-aleph conjugate of X(35)
X(35) = X(I)-beth conjugate of X(J) for these (I,J): (100,35), (643,10)


X(36) = INVERSE-IN-CIRCUMCIRCLE OF INCENTER

Trilinears       1 - 2 cos A : 1 - 2 cos B : 1 - 2 cos C
                        = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2 - bc)
                        = g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec(A/2) cos(3A/2)

Barycentrics  sin A - sin 2A : sin B - sin 2B : sin C - sin 2C
                        = g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 + c2 - a2 - bc)

If you have The Geometer's Sketchpad, you can view X(36).

X(36) is the {X(3),X(56)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(36), click More at the top of this page.

X(36) lies on these lines:
1,3   2,535   4,499   6,609   10,404   11,30   12,140   15,202   16,203   21,79   22,614   24,34   31,995   33,378   39,172   47,602   48,579   54,73   58,60   59,1110   63,997   80,104   84,90   99,350   100,519   101,672   106,901   109,953   187,1015   191,960   214,758   226,1006   238,513   255,1106   376,497   388,498   474,958   495,549   496,550   573,604   1030,1100

X(36) = midpoint of X(1) and X(484)
X(36) = reflection of X(I) in X(J) for these (I,J): (1,1319), (484,1155)
X(36) = isogonal conjugate of X(80)
X(36) = inverse-in-circumcircle of X(1)
X(36) = inverse-in-incircle of X(942)
X(36) = inverse-in-Bevan-circle of X(46)
X(36) = X(I)-Ceva conjugate of X(J) for these (I,J): (88,6), (104,1)
X(36) = crosspoint of X(58) and X(106)
X(36) = crosssum of X(I) and X(J) for these (I,J): (1,484), (10,519), (11,900)
X(36) = crossdifference of any two points on line X(37)X(650)
X(36) = X(104)-aleph conjugate of X(36)
X(36) = X(I)-beth conjugate of X(J) for these (I,J): (21,36), (100,36), (643,519)


X(37) = CROSSPOINT OF INCENTER AND CENTROID

Trilinears       b + c : c + a : a + 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 centroid of triangle AB'C', and define B" and C" cyclically. Then the lines AA", BB", CC" concur in X(37). (Eric Danneels, Hyacinthos 7892, 9/13/03)

If you have The Geometer's Sketchpad, you can view X(37).

X(37) lies on these lines:
1,6   2,75   3,975   7,241   8,941   10,594  12,225   19,25   21,172   35,267   38,354   39,596   12,225   41,584   48,205   63,940   65,71   73,836   78,965   82,251   86,190   91,498   100,111   101,284   141,742   142,1086   145,391   158,281   171,846   226,440   256,694   347,948 &n