PART 1
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 and proved 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.
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' are 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' are 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.
Possibly your browser does not recognize Greek letters; for example, pi appears on your browser as π, omega as ω, Psi as Ψ, Lambda as Λ, not equals as ≠, and intersect as ∩.
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 "Nagel" to find "Nagel point" as X(8).
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, click SKETCHES.
X(1) = INCENTER
Trilinears 1 : 1 : 1Barycentrics a : b : c
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, and is 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
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:
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 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 71,579 75,86 76,350 82,560 84,221 87,192 88,100 90,155 99,741 102,108 104,109 142,277 147,150 163,293 164,258 167,174 168,173 181,970 182,983 185,296 188,361 190,537 196,207 201,212 224,377 229,267 256,511 257,385 281,282 289,363 312,1089 320,752 321,964 329,452 335,384 336,811 341,1050 364,365 376,553 378,1063 393,836 512,875 513,764 514,663 528,1086 561,718 564,1048 572,604 573,941 607,949 631,1000 647,1021 659,891 662,897 672,1002 689,719 727,932 731,789 748,756 761,825 765,1052 908,998 1037,1041 1053,1110
X(1) = midpoint between X(I) and X(J) for these (I,J): (7,390), (8,145)
X(1) = reflection of X(I) about X(J) for these (I,J): (8,10), (40,3), (46,56), (80,11), (100,214), (191,21), (267,229), (355,5), (484,36)
X(1) = isogonal conjugate of X(1)
X(1) = isotomic conjugate of X(75)
X(1) = inverse of X(36) in the circumcircle
X(1) = inverse of X(80) in the Fuhrmann circle
X(1) = complement of X(8)
X(1) = anticomplement of X(10)
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), (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)
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), (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), (221,40), (244,513), (259,258), (266,505), (354,7), (367,366), (500,35), (513,100), (517,80), (518,291)
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) = 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)
Let X = X(1) and let V be the vector-sum XA + XB + XC; then V = X(8)X(1) = X(1)X(145).
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X;
then W = X(65)X(1) = X(8)X(72).
X(2) = CENTROID
Trilinears 1/a : 1/b : 1/c= bc : ca : ab
= csc A : csc B : csc C
Barycentrics 1 : 1 : 1
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).
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
A simple construction for * (and for square roots of points) is known:
Paul Yiu, "The uses of homogeneous barycentric coordinates in plane euclidean geometry," International Journal of Mathematical Education in Science and Technology, forthcoming.
A preprint can be downloaded from Paul Yiu's website.
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 36,535 37,75 38,244 39,76 40,946 44,89 45,88 51,262 54,68 58,540 65,959 66,206 74,113 77,189 80,214 85,241 92,273 94,300 95,97 98,110 99,111 101,116 102,117 103,118 104,119 106,121 107,122 108,123 109,124 112,127 136,925 137,930 165,516 174,236 178,188 187,316 196,653 210,354 216,232 222,651 253,1073 254,847 261,593 271,1034 254,847 261,593 271,1034 272,284 280,318 283,580 290,327 292,334 294,949 308,702 311,570 314,941 319,1100 322,1108 330,1107 351,804 355,944 366,367 371,486 372,485 392,517 476,842 495,956 496,1058 514,1022 561,716 578,1092 647,850 650,693 668,1015 670,1084 689,733 743,789 799,873
X(2) = midpoint between X(I) and X(J) for these (I,J): (3,381), (4,376), (210,354)
X(2) = reflection of X(I) about X(J) for these (I,J): (4,381), (20,376), (376,3), (381,5)
X(2) = isogonal conjugate of X(6)
X(2) = isotomic conjugate of X(2)
X(2) = cyclocevian conjugate of X(4)
X(2) = inverse of X(23) in the circumcircle
X(2) = inverse of X(858) in the nine-point circle
X(2) = inverse of X(110) in the Brocard circle
X(2) = complement of X(2)
X(2) = anticomplement of X(2)
X(2) = X(I)-Ceva conjugate of X(J) for these (I,J):
(1,192), (4,193), (6,194), (7,145), (8,144), (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)
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) = 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.
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X; then W = X(51)X(2).
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
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
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 31,601 37,975 38,976 41,218 42,967 48,71 49,155 54,97 63,72 64,154 66,141 67,542 68,343 69,332 73,212 74,110 76,98 83,262 95,264 101,103 102,109 105,277 113,122 114,127 119,123 125,131 142,516 158,243 169,910 194,385 200,963 223,1035 225,1074 238,978 252,930 256,987 269,939 296,820 298,617 299,616 302,621 303,622 315,325 352,353 388,495 390,1058 395,398 396,397 476,477 485,590 486,615 489,492 490,491 496,497 525,878 595,995 618,635 619,636 623,629 624,630 639,641 640,642 662,1098 667,1083 691,842 847,925 901,953 934,972 960,997 1037,1066 1093,1105
X(3) = midpoint between 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) about X(J) for these (I,J):
(4,5), (5,140), (6,182), (52,389), (195,54), (265,125), (355,10), (381,2), (382,4), (399,110)
X(3) = isogonal conjugate of X(4)
X(3) = isotomic conjugate of X(264)
X(3) = inverse of X(5) in the orthocentric circle
X(3) = complement of X(4)
X(3) = anticomplement 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), (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) = 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)
Let X = X(3) and let V be the vector-sum XA + XB + XC; then V = X(4)X(3) = X(382)X(4) = X(3)X(20) = X(355)X(40) = X(265)X(74) = X(52)X(185) = X(381)X(376) = X(146)X(399).
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X;
then W = X(4)X(3) = X(382)X(4) = X(3)X(20) = X(355)X(40) = X(265)X(74) = X(52)X(185) = X(381)X(376) = X(146)X(399). These are the same vectors as in the preceding list; i.e., XA + XB + XC = XA' + XB' + XC'. It is easy to prove that the unique solution X of this equation is X(3).
X(4) = ORTHOCENTER
Trilinears sec A : sec B : sec CBarycentrics tan A : tan B : tan C
The point of concurrence of the altitudes of ABC. The orthocenter 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.
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@onelist, 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 39,232 46,90 49,156 51,185 52,68 54,184 57,84 65,158 67,338 69,76 74,107 78,908 83,182 93,562 94,143 96,231 99,114 100,119 101,118 102,124 103,116 109,117 110,113 128,930 131,135 137,933 145,149 147,148 150,152 155,254 162,270 171,601 195,399 218,294 238,602 240,256 276,327 371,485 372,486 390,495 487,489 488,490 496,999 512,879 542,576 575,598 616,627 617,628 801,1092 842,935 1036,1065 1037,1067 1038,1076 1039,1096 1040,1074
X(4) = midpoint between X(I) and X(J) for these (I,J):
(3,382), (147,148), (149,153), (150,152)
X(4) = reflection of X(I) about X(J) for these (I,J):
(2,381), (3,5), (8,355), (20,3), (24,235), (40,10), (74,125), (98,115), (99,114), (100,119), (101,118), (102,124), (103,116), (104,11), (107,133), (109,117), (110,113), (112,132), (185,389), (186,403), (376,2), (378,427)
X(4) = isogonal conjugate of X(3)
X(4) = isotomic conjugate of X(69)
X(4) = cyclocevian conjugate of X(2)
X(4) = inverse of X(186) in the circumcircle
X(4) = inverse of X(403) in the nine-point circle
X(4) = complement of X(20)
X(4) = anticomplement of X(3)
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) = 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)
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)
Let X = X(4) and let V be the vector-sum XA + XB + XC; then V = X(20)X(4) = X(3)X(382).
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X;
then W = X(185)X(4) = X(52)X(382).
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) = 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
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.
X(5) lies on these lines:
1,11 2,3 6,68 10,517 13,18 14,17 32,230 33,1062 34,1060 39,114 49,54 51,52 53,216 55,498 56,499 72,908 76,262 83,98 113,125 116,118 117,124 122,133 127,132 128,137 129,130 131,136 141,211 142,971 156,184 182,206 183,315 226,912 264,1093 298,634 299,633 302,622 303,621 371,590 372,615 388,999 491,637 492,638 524,576 542,575 601,750 602,748 618,629 619,630 1090,1091
X(5) = midpoint between 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) about X(J) for these (I,J): (3,140), (52,143)
X(5) = isogonal conjugate of X(54)
X(5) = isotomic conjugate of X(95)
X(5) = inverse of X(3) in the orthocentroidal circle
X(5) = complement of X(3)
X(5) = anticomplement of X(140)
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) = X(1)-aleph conjugate of X(1048)
Let X = X(5) and let V be the vector-sum XA + XB + XC; then V = X(5)X(4) = X(3)X(5).
X(6) = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT)
Trilinears a : b : c= sin A : sin B : sin C
Barycentrics a2 : b2 : c2
The point of concurrence of the symmedians (reflections of medians about corresponding angle bisectors); the point (x, y, z), given here in actual trilinear distances, that minimizes x2 + y2 + z2.
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
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 13,14 17,18 19,34 21,941 22,251 23,353 24,54 25,51 26,143 31,42 33,204 36,609 40,380 41,48 43,87 57,222 64,185 66,427 67,125 74,112 75,239 76,83 77,241 88,89 98,262 99,729 100,739 101,106 105,1002 110,111 145,346 157,248 160,237 169,942 181,197 190,192 194,384 210,612 264,287 291,985 292,869 297,317 314,981 354,374 513,1024 517,998 519,996 523,879 561,720 598,671 603,1035 662,757 688,882 689,703 691,843 694,1084 717,789 750,899 753,825 755,827 840,919 846,1051 959,961 971,990 986,1046
X(6) = midpoint between X(69) and X(193)
X(6) = reflection of X(I) about X(J) for these (I,J): (3,182), (67,125), (69,141), (159,206)
X(6) = isogonal conjugate of X(2)
X(6) = isotomic conjugate of X(76)
X(6) = cyclocevian conjugate of X(1031)
X(6) = inverse of X(187) in the circumcircle
X(6) = inverse of X(115) in the orthocentroidal circle
X(6) = complement of X(69)
X(6) = anticomplement of X(141)
X(6) = X(I)-Ceva conjugate of X(J) for these (I,J):
(1,55), (2,3), (3,154), (4,25), (8,197), (9,198), (10,199), (54,184), (57, 56), (58,31), (68,161), (69,159), (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), (288,54), (323,399)
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) = X(I)-Hirst inverse of X(J) for these (I,J): (1,238), (2,385), (3,511), (15,16), (25,232)
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)
Let X = X(6) and let V be the vector-sum XA + XB + XC; then V = X(6)X(193) = X(69)X(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 A'A', BB', CC' concur in X(7).
X(7) lies on these lines:
1,20 2,9 3,943 4,273 6,294 8,65 11,658 21,56 27,81 37,241 33,1041 34,1039 58,272 72,443 80,150 92,189 100,1004 104,934 108,1013 109,675 171,983 174,234 177,555 190,344 192,335 193,239 218,277 225,969 253,280 256,982 274,959 281,653 286,331 310,314 354,479 513,885 517,1000 528,664 554,1082 594,599 840,927 987,1106
X(7) = reflection of X(I) about X(J) for these (I,J): (9,142), (144,9), (390,1)
X(7) = isogonal conjugate of X(55)
X(7) = isotomic conjugate of X(8)
X(7) = cyclocevian conjugate of X(7)
X(7) = complement of X(144)
X(7) = anticomplement of X(9)
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) = 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 A'A', 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.
X(8) lies on these lines:
1,2 3,100 4,72 6,594 7,65 9,346 20,40 21,55 29,219 31,987 33,1039 34,1041 35,993 37,941 38,986 56,404 58,996 76,668 79,758 80,149 81,1010 144,516 177,556 178,236 181,959 190,528 192,256 193,894 194,730 210,312 213,981 220,294 221,651 224,914 238,983 253,307 274,1002 291,330 315,760 344,480 348,664 392,1000 405,943 406,1061 442,495 443,942 474,999 475,1063 599,1086 643,1098 860,1068 908,946 1016,1083
X(8) = reflection of X(I) about X(J) for these (I,J): (1,10), (4,355), (20,40), (145,1), (149,80), (390,9)
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) = 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) = X(1)-alpeh 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)
The symmedian point of the excentral triangle.
X(9) lies on these lines:
1,6 2,7 3,84 4,10 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 58,975 100,1005 164,168 165,910 173,177 192,239 223,1073 228,1011 241,269 261,645 312,314 342,653 348,738 364,366 374,517 478,1038 498,920 522,657 607,1039 608,1041 750,896
X(9) = midpoint between X(I) and X(J) for these (I,J): (7,144), (8,390)
X(9) = reflection of X(7) about X(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), (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) = 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.
X(10) lies on these lines:
1,2 3,197 4,9 5,517 11,121 12,65 20,165 21,35 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 82,83 86,319 87,979 98,101 116,120 117,123 119,124 140,214 141,142 158,318 190,671 191,267 201,225 219,965 274,291 321,756 480,954 514,764 537,1086 626,760 631,944 775,801 894,1046 908,994
X(10) = midpoint between X(I) and X(J) for these (I,J): (1,8), (3,355), (4,40), (65,72), (80,100)
X(10) = isogonal conjugate of X(58)
X(10) = isotomic conjugate of X(86)
X(10) = complement of X(1)
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) = 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
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.
X(11) lies on these lines:
1,5 2,55 3,499 4,56 7,658 10,121 13,202 14,203 30,36 33,427 34,235 35,140 65,117 68,1069 110,215 113,942 115,1015 118,226 153,388 212,748 214,442 244,867 325,350 381,999 429,1104 518,908 523,1090
X(11) = midpoint between X(I) and X(J) for these (I,J): (1,80), (4,104), (100,149)
X(11) = reflection of X(119) about X(5)
X(11) = isogonal conjugate of X(59)
X(11) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,523), (4,513), (7,514), (8,522)
X(11) = crosspoint of X(I) and X(J) for these (I,J): (7,514), (8,522)
X(11) = X(I)-beth conjugate of X(J) for these (I,J): (11,244), (522,11), (693,11)
Let X = X(11) and let V be the vector-sum XA + XB + XC; then V = X(100)X(11) = X(11)X(149).
X(12) = HARMONIC CONJUGATE OF X(11) WRT X(1) AND X(5)
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 10,65 17,203 18,202 30,35 33,235 34,427 36,140 37,225 54,215 79,484 85,120 108,451 172,230 201,756 228,407 313,349 499,999 603,750 908,960 1091,1109
X(12) = isogonal conjugate of X(60)
X(12) = isotomic conjugate of X(261)
X(12) = X(10)-Ceva conjugate of X(201)
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*SQR(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.
X(13) lies on these lines:
2,16 3,17 4,61 5,18 6,14 11,202 15,30 76,299 98,1080 99,303 148,617 226,1082 262,383 275,472 298,532 531,671 533,621 634,635
X(13) = reflection of X(I) about X(J) for these (I,J): (14,115), (15,396)
X(13) = isogonal conjugate of X(15)
X(13) = isotomic conjugate of X(298)
X(13) = inverse of X(14) in the orthocentroidal circle
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*SQR(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.
X(14) lies on these lines:
2,15 3,18 4,62 5,17 6,13 11,203 16,30 76,298 98,383 99,302 148,616 226,554 262,1080 275,473 299,533 397,546 530,671 532,622 633,636
X(14) = reflection of X(I) about X(J) for these (I,J): (13,115), (16,395)
X(14) = isogonal conjugate of X(16)
X(14) = isotomic conjugate of X(299)
X(14) = inverse of X(13) in the orthocentroidal circle
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. The pedal triangle of X(15) is equilateral.
X(15) lies on these lines:
2,14 3,6 4,17 13,30 18,140 36,202 55,203 298,533 303,316 395,549 397,550 532,616 628,636
X(15) = reflection of X(I) about X(J) for these (I,J): (13,396), (16,187)
X(15) = isogonal conjugate of X(13)
X(15) = isotomic conjugate of X(300)
X(15) = inverse of X(16) in the circumcircle
X(15) = inverse of X(16) in Brocard circle
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) = 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.
X(16) lies on these lines:
2,13 3,6 4,18 14,30 17,140 36,203 55,202 299,532 302,316 396,549 398,550 533,617 627,635
X(16) = reflection of X(I) about X(J) for these (I,J): (14,395), (15,187)
X(16) = isogonal conjugate of X(14)
X(16) = isotomic conjugate of X(301)
X(16) = inverse of X(15) in the circumcircle
X(16) = inverse of X(15) in the Brocard
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) = 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.
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) = 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).
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) = 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
The homothetic center of the orthic and extangents triangles. Further information is available from
Paul Yiu's Website.
X(19) lies on these lines:
1,28 2,534 4,9 6,34 25,33 27,63 31,204 46,579 47,921 56,207 57,196 81,969 91,920 101,913 102,282 112,759 162,897 163,563 208,225 219,517 232,444 273,653 294,1041 604,609 960,965
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) = 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 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
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]
The reflection of X(4) about X(3); also, the orthocenter of the anticomplementary triangle.
X(20) lies on these lines:
1,7 2,3 8,40 10,165 33,1038 34,1040 55,388 56,497 57,938 58,387 64,69 68,74 72,144 78,329 98,148 99,147 100,153 101,152 103,150 104,149 109,151 110,146 145,517 155,323 185,193 391,573 393,577 394,1032 487,638 488,637 616,633 617,635 621,627 622,628 999,1058
X(20) = reflection of X(I) about X(J) for these (I,J): (2,376), (4,3), (8,40), (146,110), (147,99), (148,98), (149,104), (150,103), (151,109), (152,101), (153,100), (382,5)
X(20) = isogonal conjugate of X(64)
X(20) = isotomic conjugate of X(253)
X(20) = cyclocevian conjugate of X(1032)
X(20) = anticomplement of X(4)
X(20) = X(I)-Ceva conjugate of X(J) for these (I,J): (69,2), (489,487), (490,488)
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).
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 84,285 90,224 99,105 104,110 144,954 145,956 238,256 261,314 268,280 332,1036 612,989 614,988 741,932 748,978 884,885 915,925 976,983 1038,1041 1039,1040 1060,1063 1061,1062
X(21) = midpoint between X(1) and X(191)
X(21) = isogonal conjugate of X(65)
X(21) = anticomplement of X(422)
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) = 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)
Let X = X(21) and let V be the vector-sum XA + XB + XC; then V = X(79)X(1).
X(22) = EXETER POINT
Trilinears a(b4 + c4 - a4) : b(c4 + a4 - a4) : c(a4 + b4 - c4)Barycentrics a2(b4 + c4 - a4) : b2(c4 + a4 - a4) : c2(a4 + b4 - c4)
The perspector of the circummedial triangle and the tangential triangle; also X(22) = X(55)-of-the-tangential triangle if ABC is acute.
X(22) lies on these lines:
2,3 6,251 35,612 36,614 51,182 56,977 69,159 98,925 99,305 100,197 110,154 157,183 160,325 161,343 184,511 232,577
X(22) = reflection of X(378) about X(3)
X(22) = isogonal conjugate of X(66)
X(22) = inverse of X(858) in the circumcircle
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) = 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)
The inverse of the centroid in the circumcircle.
X(23) lies on these lines:
2,3 6,353 51,575 94,98 110,323 111,187 159,193 184,576 232,250 385,523
X(23) = reflection of X(323) about X(110)
X(23) = isogonal conjugate of X(67)
X(23) = inverse of X(2) in the circumcircle
X(23) = anticomplement of X(427)
X(23) = crosspoint of X(111) and X(251)
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.
X(24) lies on these lines:
2,3 6,54 32,232 33,35 34,36 49,568 51,578 64,74 96,847 107,1093 108,915 110,155 184,389 254,393 511,1092
X(24) = reflection of X(4) about X(235)
X(24) = isogonal conjugate of X(68)
X(24) = inverse of X(403) in the circumcircle
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) = 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.
X(25) lies on these lines:
1,1036 2,3 6,51 19,33 31,608 34,56 41,42 52,155 53,157 58,967 92,242 98,107 105,108 111,112 114,135 132,136 143,156 183,264 185,1078 262,275 317,325 371,493 372,494 393,1033 394,511 669,878 692,913
X(25) = isogonal conjugate of X(69)
X(25) = isotomic conjugate of X(305)
X(25) = inverse of X(468) in the circumcircle
X(25) = inverse of X(427) in the orthocentroidal circle
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) = 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]Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2cos 2B + c2cos 2C - a2cos 2A)
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 68,161 154,155 206,511
X(26) = reflection of X(155) about X(156)
X(26) = isogonal conjugate of X(70)
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)
X(27) lies on these lines:
2,3 7,81 19,63 57,273 58,270 103,107 110,917 226,284 295,335 306,1043 393,967 648,903 662,913
X(27) = isogonal conjugate of X(71)
X(27) = isotomic conjugate of X(306)
X(27) = inverse of X(469) in the orthocentroidal circle
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) = 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)
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)
X(28) lies on these lines:
1,19 2,3 33,975 34,57 56,278 60,81 88,162 104,107 105,112 108,225 110,915 228,943 242,261 272,273 279,1014 281,958 607,1002 608,959
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) = 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)
X(29) lies on these lines:
1,92 2,3 8,219 33,78 34,77 58,162 65,296 81,189 102,107 226,951 242,257 270,283 284,950 314,1039 388,1037 497,1036 515,947 1056,1059 1057,1058
X(29) = isogonal conjugate of X(73)
X(29) = isotomic conjugate of X(307)
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) = 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)
The point of intersection of the Euler line and the line at infinity. Thus, each of the 22 lines listed below is parallel to the Euler line.
X(30) lies on these lines:
1,79 2,3 11,36 12,35 13,15 14,16 33,1060 34,1062 40,191 52,185 53,577 55,495 56,496 61,397 62,398 64,68 74,265 80,484 98,671 99,316 110,477 115,187 143,389 146,323 148,385 155,1078 182,597 262,598 298,616 299,617 390,1056 489,638 490,637 497,999 511,512 551,946 553,942 618,623 619,624 620,625 944,962
X(30) = isogonal conjugate of X(74)
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(31) = 2nd POWER POINT
Trilinears a2 : b2 : c2Barycentrics a3 : b3 : c3
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 92,162 101,609 110,593 163,923 184,604 etc.
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) = 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)
Barycentrics a4 : b4 : c4
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) = midpoint between X(371) and X(372)
X(32) = isogonal conjugate of X(76)
X(32) = inverse of X(39) in the Brocard circle
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) = 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(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)
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) = 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) = X(33)-beth conjugate of X(25)
X(34)
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)
The center of perspective of the orthic triangle and the reflection about the incenter of the intangents triangle.
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) = 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) = 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(35)
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)
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 etc.
X(35) = isogonal conjugate of X(79)
X(35) = inverse of X(484) in the circumcircle
X(35) = X(500)-cross conjugate of X(1)
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 OF THE INCENTER IN THE CIRCUMCIRCLE
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 cos2(A/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)
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 between X(1) and X(484)
X(36) = isogonal conjugate of X(80)
X(36) = inverse of X(1) in the circumcircle
X(36) = inverse of X(942) in the incircle>BR>
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) = 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)
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) = midpoint between 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) = 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) = 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)
Let X = X(37) and let V be the vector-sum XA + XB + XC; then V = X(75)X(37) = X(37)X(192).
X(38)
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) = isogonal conjugate of X(82)
X(38) = crosspoint of X(1) and X(75)
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)
Barycentrics a2(b2 + c2) : b2(c2 + a2) : c2(a2 + b2)
The midpoint between the 1st and 2nd Brocard points, given by trilinears c/b : a/c : b/a and b/c : c/a : a/b .
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
Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 10: The Brocard Points.
X(39) = midpoint between X(76) and X(194)
X(39) = isogonal conjugate of X(83)
X(39) = isotomic conjugate of X(308)
X(39) = inverse of X(32) in the Brocard circle
X(39) = complement of X(76)
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)
Let X = X(39) and let V be the vector-sum XA + XB + XC; then V = X(76)X(39) = X(39)X(194).
X(40) = REFLECTION OF THE INCENTER IN CIRCUMCENTER
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)
The point of concurrence of the perpendiculars from the excenters to the respective sides; also, the circumcenter of the excentral triangle.
X(40) lies on 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
X(40) = midpoint between X(8) and X(20)
X(40) = reflection of X(I) about X(J) for these (I,J): (1,3), (4,10)
X(40) = isogonal conjugate of X(84)
X(40) = isotomic conjugate of X(309)
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), (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) = 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), (40,40), (643,78), (644,728)
X(41)
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)
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) = 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) = 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)
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) = 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) = 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
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) = 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) = 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) = midpoint between X(190) and X(239)
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) = 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)
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) = isogonal conjugate of X(89)
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) = reflection of X(1) about X(56)
X(46) = isogonal conjugate of X(90)
X(46) = X(4)-Ceva conjugate of X(1)
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)
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) = 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) = 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(48)
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) = isogonal conjugate of X(92)
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) = 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) = 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)
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) = isogonal conjugate of X(94)
X(50) = inverse of X(566) in the Brocard circle
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(51) = CENTROID OF THE 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) = reflection of X(210) about X(375)
X(51) = isogonal conjugate of X(95)
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)
Let X = X(51) and let V be the vector-sum XA + XB + XC; then V = X(3)X(52) = X(20)X(185)
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X; then W = X(428)X(51).
X(52) = ORTHOCENTER OF THE 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) = reflection of X(I) about X(J) for these (I,J): (3,389), (5,143)
X(52) = isogonal conjugate of X(96)
X(52) = inverse of X(569) in the Brocard circle
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(53) = SYMMEDIAN POINT OF THE 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) = 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(54) = KOSNITA POINT
Trilinears sec(B - C) : sec(C - A) : sec(A - B)Barycentrics tan(B - C) : tan(C - A) : tan(A - B)
John Rigby, "Brief notes on some forgotten geometrical theorems," Mathematics and Informatics Quarterly 7 (1997) 156-158.
X(54) lies on these lines:
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 74,185 112,217 140,252 156,381 186,389 276,290 575,895 826,879
X(54) = midpoint between X(3) and X(195)
X(54) = isogonal conjugate of X(5)
X(54) = isotomic conjugate of X(311)
X(54) = X(I)-Ceva conjugate of X(J) for these (I,J): (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(55) = INTERNAL CENTER OF SIMILITUDE OF CIRCUMCIRCLE AND 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)
The center of homothety of three triangles: tangential, intangents, and extangents.
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) = isogonal conjugate of X(7)
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) = X(43)-Hirst inverse of X(238)
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) = EXTERNAL CENTER OF SIMILITUDE OF CIRCUMCIRCLE AND 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 about 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) = midpoint between X(1) and X(46)
X(56) = isogonal conjugate of X(8)
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) = X(266)-aleph conjugate of X(1050)
X(57) = 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) lies on 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 92,653 164,177 169,277 173,174 200,518 201,975 234,362 239,330 255,580 279,479 345,728 497,516 499,920 649,1024 658,673 748,896 758,997 955,991 957,995 959,1042 961,1106 978,1046 1020,1086
X(57) = isogonal conjugate of X(9)
X(57) = isotomic conjugate of X(312)
X(57) = complement of X(329)
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) = X(1)-Hirst inverse of X(241)
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) 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) = isogonal conjugate of X(10)
X(58) = isotomic conjugate of X(313)
X(58) = inverse of X(386) in the Brocard circle
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) = 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)
X(59) = X(765)-beth conjugate of X(765)
X(60)
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)
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 these lines:
1,203 2,18 3,6 4,13 5,14 30,397 56,202 140,395 299,636 302,629 618,627
X(61) = isogonal conjugate of X(17)
X(61) = inverse of X(62) in the Brocard circle
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)
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X = X(61); then W = X(397)X(61).
X(62)
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 on these lines:
1,202 2,17 3,6 4,14 5,13 30,398 56,203 140,396 298,635 303,630 619,628
X(62) = isogonal conjugate of X(18)
X(62) = inverse of X(61) in the Brocard circle
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)
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X = X(62); then W = X(398)X(62).
X(63)
Trilinears cot A : cot B : cot C= b2 + c2 - a2 : c2 + a2 - b2 : c2 + 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) = isogonal conjugate of X(19)
X(63) = isotomic conjugate of X(92)
X(63) = anticomplement of X(226)
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) = 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)
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)
X(64) lies on these lines:
3,154 6,185 20,69 24,74 30,68 33,65 40,72 54,378 55,73 71,198 265,382
X(64) = isogonal conjugate of X(20)
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)
The perspector of ABC and the Yff central 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) = reflection of X(72) about X(10)
X(65) = isogonal conjugate of X(21)
X(65) = isotomic conjugate of X(314)
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) = 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)
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) = reflection of X(159) about X(141)
X(66) = isogonal conjugate of X(22)
X(66) = isotomic conjugate of X(315)
X(66) = anticomplement of X(206)
X(66) = cevapoint of X(125) and X(512)
X(66) = X(32)-cross conjugate of X(2)
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X = X(66); then W = X(185)X(64).
X(67)
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) = reflection of X(I) about 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) = cevapoint of X(141) and X(524)
X(67) = X(187)-cross conjugate of X(2)
X(68)
Trilinears cos A sec 2A : cos B sec 2B : cos C sec 2CBarycentrics tan 2A : tan 2B : tan 2C
X(68) lies on these lines:
2,54 3,343 4,52 5,6 11,1069 20,74 26,161