PART 2
X(201) = X(10)-CEVA CONJUGATE OF X(12)
Trilinears (cos A)[1 + cos(B - C)] : (cos B)[1 + cos(C - A)] : (cos C)[1 + cos(A - B)]Barycentrics (sin 2A)[1 + cos(B - C)] : (sin 2B)[1 + cos(C - A)] : (sin 2C)[1 + cos(A - B)]
X(201) lies on these lines:
1,212 9,34 10,225 12,756 33,40 37,65 38,56 55,774 57,975 63,603 72,73 109,191 210,227 220,221 255,1060 337,348 388,984 601,920
X(201) = isogonal conjugate of X(270)
X(201) = X(10)-Ceva conjugate of X(12)
X(201) = crosspoint of X(10) and X(72)
X(201) = X(I)-beth conjugate of X(J) for these (I,J): (72,201), (1018,201)
X(202) = X(1)-CEVA CONJUGATE OF X(15)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = u(v + w - u),u = u(A,B,C) = sin(A + π/3), v = u(B,C,A), w = u(C,A,B)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(202) lies on these lines:
1,62 6,101 11,13 12,18 15,36 16,55 17,499 56,61 395,495 397,496
X(202) = X(1)-Ceva conjugate of X(15)
X(203) = X(1)-CEVA CONJUGATE OF X(16)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = u(v + w - u),u = u(A,B,C) = sin(A - π/3), v = u(B,C,A), w = u(C,A,B)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(203) lies on these lines:
1,61 6,101 11,14 12,17 15,55 16,36 18,499 56,62 396,495 398,496
X(203) = X(1)-Ceva conjugate of X(16)
X(204) = X(1)-CEVA CONJUGATE OF X(19)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan A)(tan B + tan C - tan A)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(204) lies on these lines: 6,33 19,31 25,34 55,1033 63,162 108,223 207,221
X(204) = X(1)-Ceva conjugate of X(19)
X(204) = X(I)-beth conjugate of X(J) for these (I,J): (108,204), (162,223)
X(205) = X(9)-CEVA CONJUGATE OF X(31)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2[b2tan B/2 + c2tan C/2 - a2tan A/2]Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(205) lies on these lines: 25,41 37,48 78,101 154,220 184,213
X(205) = X(9)-Ceva conjugate of X(31)
X(206) = X(2)-CEVA CONJUGATE OF X(32)
Trilinears a3(b4 + c4 - a4) : b3(c4 + a4 - b4) : c3(a4 + b4 - c4)Barycentrics a4(b4 + c4 - a4) : b4(c4 + a4 - b4) : c4(a4 + b4 - c4)
This is also X(66) of the medial triangle.
X(206) lies on these lines:
2,66 5,182 6,25 26,511 69,110 157,216 160,577 219,692 237,571
X(206) = midpoint between X(6) and X(159)
X(206) = complement of X(66)
X(206) = X(2)-Ceva conjugate of X(32)
X(206) = crosspoint of X(2) and X(315)
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(206); then W = X(66)X(141) = X(141)X(159).
X(207) = X(1)-CEVA CONJUGATE OF X(34)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 - sec A)(sec B + sec C - sec A - 1)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(207) lies on these lines: 1,196 19,56 33,64 34,1042 40,108 78,653 204,221
X(207) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,34), (196,19)
X(207) = X(1)-beth conjugate of X(64)
X(208) = X(4)-CEVA CONJUGATE OF X(34)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 - sec A)(cos B + cos C - cos A - 1)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(208) lies on these lines:
1,102 4,57 19,225 25,34 33,64 40,196 198,227 226,406 318,653
X(208) = isogonal conjugate of X(271)
X(208) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,34), (57,19), (342,223)
X(208) = X(I)-beth conjugate of X(J) for these (I,J): (108,208), (162,1)
X(209) = X(4)-CEVA CONJUGATE OF X(37)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin B + sin C)[sin A + sin(A - B) + sin(A - C)]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(209) lies on these lines: 6,31 10,12 44,51 306,518
X(209) = isogonal conjugate of X(272)
X(209) = X(4)-Ceva conjugate of X(37)
X(210) = X(10)-CEVA CONJUGATE OF X(37)
Trilinears (b + c)(b + c - a) : (c + a)(c + a - b) : (a + b)(a + b - c)Barycentrics a(b + c)(b + c - a) : b(c + a)(c + a - b) : c(a + b)(a + b - c)
X(210) lies on these lines:
2,354 6,612 8,312 9,55 10,12 31,44 33,220 37,42 38,899 43,984 45,968 51,374 56,936 63,1004 78,958 165,971 201,227 213,762 381,517 392,519 430,594 869,1107 956,997 976,1104
X(210) = reflection of X(I) about X(J) for these (I,J): (51,375), (354,2)
X(210) = isogonal conjugate of X(1014)
X(210) = X(10)-Ceva conjugate of X(37)
X(210) = crosspoint of X(8) and X(9)
X(210) = X(I)-beth conjugate of X(J) for these (I,J): (200,210), (210,42)
Let X = X(210) and let V be the vector-sum XA + XB + XC; then V = X(65)X(8) = X(1)X(72).
X(211) = X(4)-CEVA CONJUGATE OF X(39)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C)= sin(A + ω)[cos B sin(B + ω) + cos C sin(C + ω) - cos A sin(A + ω)]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(211) lies on these lines: 5,141 32,184 52,114
X(211) = X(4)-Ceva conjugate of X(39)
X(212) = X(9)-CEVA CONJUGATE OF X(41)
Trilinears (cos A)(1 + cos A) : (cos B)(1 + cos B) : (cos C)(1 + cos C)= (cos A)cos2(A/2) : (cos B)cos2(B/2) : (cos C)cos2(C/2)
Barycentrics (sin 2A)(1 + cos A) : (sin 2B)(1 + cos B) : (sin 2C)(1 + cos C)
X(212) lies on these lines:
1,201 3,73 6,31 9,33 11,748 34,40 35,47 48,184 56,939 63,1040 78,283 109,165 154,198 238,497 312,643 582,942
X(212) = isogonal conjugate of X(273)
X(212) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,48), (9,41), (283,219)
X(212) = X(228)-cross conjugate of X(55)
X(212) = crosspoint of X(I) and X(J) for these (I,J): (3,219), (9,78)
X(212) = X(212)-beth conjugate of X(184)
X(213) = X(6)-CEVA CONJUGATE OF X(42)
Trilinears (b + c)a2 : (c + a)b2 : (a + b)c2Barycentrics (b + c)a3 : (c + a)b3 : (a + b)c3
X(213) lies on these lines: 1,6 8,981 31,32 39,672 58,101 63,980 83,239 100,729 184,205 274,894 607,1096 667,875 692,923
X(213) = isogonal conjugate of X(274)
X(213) = X(I)-Ceva conjugate of X(J) for these (I,J): (6,42), (37,228)
X(213) = crosspoint of X(6) and X(31)
X(213) = X(I)-beth conjugate of X(J) for these (I,J): (41,213), (101,65), (644,213)
X(214) = X(2)-CEVA CONJUGATE OF X(44)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - 2a)(b2 + c2 - a2 - bc)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(214) lies on these lines: 1,88 2,80 9,48 10,140 11,442 36,758 44,1017 119,515 142,528 535,908 662,759 1015,1100
X(214) = midpoint between X(1) and X(100)
X(214) = complement of X(80)
X(214) = X(2)-Ceva conjugate of X(44)
X(214) = crosspoint of X(2) and X(320)
X(214) = X(21)-beth conjugate of X(244)
X(215) = X(1)-CEVA CONJUGATE OF X(50)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin 3A)(sin 3B + sin 3C - sin 3A)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(215) lies on these lines: 1,49 11,110 12,54 55,184
X(215) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,50)
X(216) = X(5)-CEVA CONJUGATE OF X(51)
Trilinears sin 2A cos(B - C) : sin 2B cos(C - A) : sin 2C cos(A - B)Barycentrics (sin A)(sin 2A)cos(B - C) : sin B sin 2B cos(C - A) : sin C sin 2C cos(A - B)
X(216) lies on these lines:
2,232 3,6 5,53 51,418 95,648 97,288 115,131 157,206 395,465 373,852 395,465 396,466 631,1075 1015,1060
X(216) = isogonal conjugate of X(275)
X(216) = isotomic conjugate of X(276)
X(216) = inverse of X(577) in the Brocard circle
X(216) = complement of X(264)
X(216) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,5), (3,418), (5,51), (324,52)
X(216) = cevapoint of X(217) and X(418)
X(216) = X(217)-cross conjugate of X(51)
X(216) = crosspoint of X(I) and X(J) for these (I,J): (2,3), (5,343)
X(217) = X(6)-CEVA CONJUGATE OF X(51)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin3A) cos A cos(B - C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(217) lies on these lines: 4,6 32,184 39,185 54,112 83,287 232,389
X(217) = isogonal conjugate of X(276)
X(217) = X(I)-Ceva conjugate of X(J) for these (I,J): (6,51), (216,418)
X(217) = crosspoint of X(I) and X(J) for these (I,J): (6,184), (51,216)
X(218) = X(7)-CEVA CONJUGATE OF X(55)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = cos2(A/2) [cos4(B/2) + cos4(C/2)- cos4(A/2)]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(218) lies on these lines:
1,6 3,41 4,294 7,277 32,906 43,170 46,910 56,101 65,169 145,644 198,579 222,241 279,651
X(218) = isogonal conjugate of X(277)
X(218) = eigencenter of cevian triangle of X(7)
X(218) = eigencenter of anticevian triangle of X(55)
X(218) = X(7)-Ceva conjugate of X(55)
X(218) = X(644)-beth conjugate of X(218)
X(219) = X(8)-CEVA CONJUGATE OF X(55)
Trilinears cos A cos A/2 : cos B cos B/2 : cos C cos C/2= (sin A)/(1 - sec A) : (sin B)/(1 - sec B) : (sin C)/(1 - sec C)
= 1/(csc A - 2 csc 2A) : 1/(csc B - 2 csc 2B) : 1/(csc C - 2 csc 2C)
Barycentrics sin 2A cos A/2 : sin 2B cos B/2 : sin 2C cos C/2
X(219) lies on these lines:
1,6 3,48 8,29 10,965 19,517 40,610 41,1036 55,284 56,579 63,77 101,102 144,347 200,282 206,692 255,268 278,329 332,345 346,644 572,947 577,906 604,672
X(219) = isogonal conjugate of X(278)
X(219) = isotomic conjugate of X(331)
X(219) = X(I)-Ceva conjugate of X(J) for these (I,J): (8,55), (63,3), (283,212)
X(219) = X(I)-cross conjugate of X(J) for these (I,J): (48,268), (71,9), (212,3)
X(219) = crosspoint of X(I) and X(J) for these (I,J): (8,345), (64,78)
X(219) = X(I)-beth conjugate of X(J) for these (I,J): (101,478), (219,48), (644,219)
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(219); then W = X(19)X(219).
X(220) = X(9)-CEVA CONJUGATE OF X(55)
Trilinears csc A/2 cos3(A/2) : csc B/2 cos3(B/2) : csc C/2 cos3(C/2)Barycentrics sin A csc A/2 cos3(A/2) : sin B csc B/2 cos3(B/2) : sin C csc C/2 cos3(C/2)
X(220) lies on these lines:
1,6 3,101 8,294 33,210 40,910 41,55 48,963 63,241 64,71 78,949 144,279 154,205 169,517 200,728 201,221 268,577 277,1086 281,594 329,948 346,1043
X(220) = isogonal conjugate of X(279)
X(220) = X(I)-Ceva conjugate of X(J) for these (I,J): (9,55), (200,480)
X(220) = cevapoint of X(1) and X(170)
X(220) = crosspoint of X(9) and X(200)
X(220) = X(I)-beth conjugate of X(J) for these (I,J): (101,221), (220,41), (644,220), (728,728)
X(221) = X(1)-CEVA CONJUGATE OF X(56)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin2A/2)(cos B + cos C - cos A - 1)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(221) lies on these lines:
1,84 3,102 6,19 8,651 31,56 40,223 55,64 201,220 204,207 960,1038
X(221) = isogonal conjugate of X(280)
X(221) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,56), (222,6), (223,198)
X(221) = crosspoint of X(I) and X(J) for these (I,J): (1,40), (196,347)
X(221) = X(I)-beth conjugate of X(J) for these (I,J): (1,34), (40,40), (101,220), (109,221), (110,3)
X(222) = X(7)-CEVA CONJUGATE OF X(56)
Trilinears cos A tan A/2 : cos B tan B/2 : cos C tan C/2= 1/(csc A + 2 csc 2A) : 1/(csc B + 2 csc 2B) : 1/(csc A + 2 csc 2C)
= a(b2 + c2 - a2)/(b + c - a) : b(c2 + a2 - b2)/(c + a - b) : c(a2 + b2 - c2)/(a + b - c)
Barycentrics a2/(1 + sec A) : b2/(1 + sec B) : c2/(1 + sec C)
X(222) lies on these lines:
1,84 2,651 3,73 6,57 7,27 33,971 34,942 46,227 55,103 56,58 63,77 72,1038 171,611 189,281 218,241 226,478 268,1073 581,1035 601,1066 613,982 912,1060
X(222) = isogonal conjugate of X(281)
X(222) = X(I)-Ceva conjugate of X(J) for these (I,J): (7,56), (77,3), (81,57)
X(222) = cevapoint of X(6) and X(221)
X(222) = X(I)-cross conjugate of X(J) for these (I,J): (48,3), (73,77)
X(222) = crosspoint of X(7) and X(348)
X(222) = X(I)-beth conjugate of X(J) for these (I,J):
(21,1012), (63,63), (110,222), (287,222), (648,222), (651,222), (662,2), (895,222)
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(222); then W = X(33)X(222).
X(223) = X(2)-CEVA CONJUGATE OF X(57)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan A/2)(cos B + cos C - cos A - 1]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(223) lies on these lines:
1,4 2,77 3,1035 6,57 9,1073 40,221 56,937 63,651 108,204 109,165 312,664 329,347 380,608 580,603 936,1038
X(223) = isogonal conjugate of X(282)>BR>
X(223) = complement of X(189)
X(223) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,57), (77,1), (342,208), (347,40)
X(223) = cevapoint of X(198) and X(221)
X(223) = X(I)-cross conjugate of X(J) for these (I,J): (198,40), (227,347)
X(223) = crosspoint of X(2) and X(329)
X(223) = X(I)-aleph conjugate of X(J) for these (I,J):
(63,1079), (77,223), (81,580), (174,46), (508,19), (651,109)
X(223) = X(I)-beth conjugate of X(J) for these (I,J):
(2,278), (100,200), (162,204), (329,329), (651,223), (662,63)
X(224) = X(7)-CEVA CONJUGATE OF X(63)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = [cot B cos2(B/2) + cot C (cot C/2)2 - cot A (cot C/2)2]cot A
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(224) lies on these lines: 1,377 3,63 8,914 21,90 46,100 65,1004
X(224) = X(7)-Ceva conjugate of X(63)
X(225) = X(4)-CEVA CONJUGATE OF X(65)
Trilinears (sec A)(cos B + cos C) : (sec B)(cos C + cos A) : (sec C)(cos A + cos B)Barycentrics (tan A)(cos B + cos C) : (tan B)(cos C + cos A) : (tan C)(cos A + cos B)
X(225) lies on these lines:
1,4 3,1074 7,969 10,201 12,37 19,208 28,108 46,254 65,407 75,264 91,847 158,1093 377,1038 412,775 653,897
X(225) = isogonal conjugate of X(283)
X(225) = isotomic conjugate of X(332)
X(225) = X(4)-Ceva conjugate of X(65)
X(225) = X(407)-cross conjugate of X(4)
X(225) = crosspoint of X(I) and X(J) for these (I,J): (4,158), (273,278)
X(225) = X(I)-beth conjugate of X(J) for these (I,J): (4,225), (10,227), (108,1042), (318,10)
X(226) = X(7)-CEVA CONJUGATE OF X(65)
Trilinears (csc A)(cos B + cos C) : (csc B)(cos C + cos A) : (csc C)(cos A + cos B)= bc(b + c)/(b + c - a) : ca(c + a)/(c + a - b) : ab(a + b)/(a + b - c)
Barycentrics (b + c)/(b + c - a) : (c + a)/(c + a - b) : (a + b)/(a + b - c)
This center is also X(63) of the medial triangle.
X(226) lies on these lines:
1,4 2,7 5,912 10,12 11,118 13,1082 14,554 27,284 29,951 35,79 36,1006 37,440 41,379 46,498 55,516 56,405 76,85 78,377 81,651 92,342 98,109 102,1065 196,281 208,406 222,478 228,851 262,982 273,469 306,321 443,936 481,485 482,486 495,517 535,551 664,671 975,1038 990,1040
X(226) = isogonal conjugate of X(284)
X(226) = isotomic conjugate of X(333)
X(226) = complement of X(63)
X(226) = X(I)-Ceva conjugate of X(J) for these (I,J): (7,65), (349,307)
X(226) = cevapoint of X(37) and X(65)
X(226) = X(I)-cross conjugate of X(J) for these (I,J): (37,10), (73,307)
X(226) = crosspoint of X(2) and X(92)
X(226) = X(I)-beth conjugate of X(J) for these (I,J): (2,226), (21,1064), (100,42), (190,226), (312,306), (321,321), (335,226), (835,226)
X(227) = X(10)-CEVA CONJUGATE OF X(65)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(cos B + cos C - cos A - 1)tan 2ABarycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(227) lies on these lines:
12,37 34,55 40,221 42,65 46,222 56,197 198,208 201,210 322,347 607,910
X(227) = isogonal conjugate of X(285)
X(227) = X(10)-Ceva conjugate of X(65)
X(227) = crosspoint of X(223) and X(347)
X(227) = X(I)-beth conjugate of X(J) for these (I,J): (10,225), (40,227), (100,72)
X(228) = X(3)-CEVA CONJUGATE OF X(71)
Trilinears (sin 2A)(sin B + sin C) : (sin 2B)(sin C + sin A) : (sin 2C)(sin A + sin B)Barycentrics (sin A sin 2A)(sin B + sin C) : (sin B sin 2B)(sin C + sin A) : (sin C sin 2C)(sin A + sin B)
X(228) lies on these lines:
3,63 9,1011 12,407 19,25 28,943 31,32 35,846 42,181 48,184 73,408 98,100 226,851
X(228) = isogonal conjugate of X(286)
X(228) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,71), (37,213), (55,42)
X(228) = crosspoint of X(I) and X(J) for these (I,J): (3,48), (37,72), (55,212), (71,73)
X(228) = X(212)-beth conjugate of X(228)
X(229) = X(7)-CEVA CONJUGATE OF X(81)
Trilinears f(a,b,c) : f(b,c,a : f(c,a,b), where f(a,b,c) = (v + w - u)/(b + c),u = u(a,b,c) = a(b + c - a)/(b + c), v = u(b,c,a), w = u(c,a,b)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(229) lies on these lines: 1,267 21,36 28,60 58,244 65,110 593,1104
X(229) = midpoint between X(1) and X(267)
X(229) = X(7)-Ceva conjugate of X(81)
X(230) = X(2)-CEVA CONJUGATE OF X(114)
Trilinears f(a,b,c) : f(b,c,a : f(c,a,b), wheref(a,b,c) = bc[a2(2a2 - b2 - c2) + (b2 - c2)2]
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(230) lies on these lines:
2,6 5,32 12,172 25,53 30,115 39,140 50,858 111,476 112,403 231,232 393,459 538,620 549,574 625,754
X(230) = midpoint between X(I) and X(J) for these (I,J): (115,187), (325,385), (395,396)
X(230) = complement of X(325)
X(230) = X(2)-Ceva conjugate of X(114)
X(230) = crosspoint of X(2) and X(98)
X(230) = X(2)-Hirst inverse of X(193)
X(230) = X(I)-beth conjugate of X(J) for these (I,J): (281,230), (645,230)
Let X = X(230) and let V be the vector-sum XA + XB + XC; then V = X(230)X(385) = X(265)X(399).
X(231) = X(2)-CEVA CONJUGATE OF X(128)
Trilinears f(a,b,c) : f(b,c,a : f(c,a,b), where f(a,b,c) = u(-au + bv + cw), u : v : w = X(128)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(231) lies on these lines: 4,96 6,17 50,115 230,232
X(231) = X(2)-Ceva conjugate of X(128)
X(231) = X(281)-beth conjugate of X(230)
X(232) = X(2)-CEVA CONJUGATE OF X(132)
Trilinears tan A cos(A + ω) : tan B cos(B + ω) : tan C cos(C = ω)Barycentrics sin A tan A cos(A + ω) : sin B tan B cos(B + ω) : sin C tan C cos(C = ω)
X(232) lies on these lines:
2,216 4,39 6,25 19,444 22,577 23,250 24,32 53,427 112,186 115,403 217,389 230,231 297,325 378,574 385,648 459,800
X(232) = isogonal conjugate of X(287)
X(232) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,132), (297,511)
X(232) = X(237)-cross conjugate of X(511)
X(232) = X(6)-Hirst inverse of X(25)
X(232) = X(281)-beth conjugate of X(232)
X(233) = X(2)-CEVA CONJUGATE OF X(140)
Trilinears f(a,b,c) : f(b,c,a : f(c,a,b), where f(a,b,c) = [b cos(C - A) + c cos(B - A)]cos(B - C)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(233) lies on these lines: 2,95 5,53 6,17 115,128 122,138
X(233) = isogonal conjugate of X(288)
X(233) = complement of X(95)
X(233) = X(2)-Ceva conjugate of X(140)
X(233) = crosspoint of X(2) and X(5)
X(234) = X(7)-CEVA CONJUGATE OF X(177)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos B/2 + cos C/2)(cos B/2 cos C/2)2Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(234) lies on these lines: 2,178 7,174 57,362 75,556 555,1088
X(234) = X(7)-Ceva conjugate of X(177)
X(235) = X(4)-CEVA CONJUGATE OF X(185)
Trilinears f(a,b,c) : f(b,c,a : f(c,a,b), where f(a,b,c) = u(-u cos A + v cos B + w cos C), where u : v : w = X(185)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(235) lies on these lines: 2,3 11,34 12,33 52,113 133,136
X(235) = midpoint between X(4) and X(24)
X(235) = X(4)-Ceva conjugate of X(185)
X(236) = X(2)-CEVA CONJUGATE OF X(188)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A/2)(cos B/2 + cos C/2 - cos A/2)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(236) lies on these lines: 2,174 8,178 9,173
X(236) = isogonal conjugate of X(289)
X(236) = X(2)-Ceva conjugate of X(188)
Centers 237- 248
are line conjugates. The P-line conjugate of Q is the point
where line PQ meets the polar of the isogonal conjugate of Q.
X(237) = X(3)-LINE CONJUGATE OF X(2)
Trilinears a2cos(A + ω) : b2cos(B + ω)2cos(C + ω)Barycentrics a3cos(A + ω) : b3cos(B + ω) : c3cos(C + ω)
X(237) lies on these lines: 2,3 6,160 31,904 32,184 39,51 154,682 187,351 206,571
X(237) = isogonal conjugate of X(290)
X(237) = X(98)-Ceva conjugate of X(6)
X(237) = crosspoint of X(I) and X(J) for these (I,J): (6,98), (232,511)
X(237) = X(32)-Hirst inverse of X(184)
X(237) = X(3)-line conjugate of X(2)
X(237) = X(55)-beth conjugate of X(237)
X(238) = X(1)-LINE CONJUGATE OF X(37)
Trilinears a2 - bc : b2 - ca : c2 - abBarycentrics a3 - abc : b3 - abc : c3 - abc
X(238) lies on these lines:
1,6 2,31 3,978 4,602 8,983 10,82 21,256 36,513 43,55 47,499 56,87 58,86 63,614 100,899 105,291 106,898 162,415 190,726 212,497 239,740 242,419 244,896 516,673 517,1052 519,765 580,946 601,631 942,1046 992,1009 993,995 1006,1064
X(238) = isogonal conjugate of X(291)
X(238) = isotomic conjugate of X(334)
X(238) = X(I)-Ceva conjugate of X(J) for these (I,J): (105,1), (292,171)
X(238) = X(I)-Hirst inverse of X(J) for these (I,J): (1,6), (43,55)
X(238) = X(1)-line conjugate of X(37)
X(238) = X(105)-aleph conjugate of X(238)
X(238) = X(I)-beth conjugate of X(J) for these (I,J): (21,238), (643,902), (644,238), (932,238)
Let X = X(238) and let V be the vector-sum XA + XB + XC; then V = X(320)X(1).
X(239) = X(1)-LINE CONJUGATE OF X(42)
Trilinears bc(a2 - bc) : ca(b2 - ca) : ab(c2 - ab)Barycentrics a2 - bc : b2 - ca : c2 - ab
X(239) lies on these lines:
1,2 6,75 7,193 9,192 44,190 57,330 63,194 81,274 83,213 86,1100 92,607 141,319 238,740 241,664 257,333 294,666 318,458 320,524 335,518 514,649 1043,1104
X(239) = isogonal conjugate of X(292) = isotomic conjugate of X(335)
X(239) = reflection of X(190) about X(44)
X(239) = crosspoint of X(256) and X(291)
X(239) = X(I)-Hirst inverse of X(J) for these (I,J): (1,2), (9,192)
X(239) = X(1)-line conjugate of X(42)
X(239) = X(I)-beth conjugate of X(J) for these (I,J): (333,239), (645,44)
X(240) = X(1)-LINE CONJUGATE OF X(48)
Trilinears sec A cos(A + ω) : sec B cos(B + ω) : sec C cos(C + ω)Barycentrics tan A cos(A + ω) : tan B cos(B + ω) : tan C cos(C + ω)
X(240) lies on these lines: 1,19 4,256 38,92 63,1096 75,158 162,896 278,982 281,984 522,656 607,611 608,613
X(240) = isogonal conjugate of X(293)
X(240) = isotomic conjugate of X(336)
X(240) = X(1)-Hirst inverse of X(19)
X(240) = X(31-line conjugate of X(48)
X(240) = X(318)-beth conjugate of X(240)
X(241) = X(1)-LINE CONJUGATE OF X(55)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = cos4B/2 - [cos2(A/2)][cos2(B/2) +cos2(C/2)] + cos4(C/2)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(241) lies on these lines: 1,3 2,85 6,77 7,37 9,269 44,651 63,220 141,307 218,222 239,664 277,278 294,910 347,1108 514,650 960,1042
X(241) = isogonal conjugate of X(294)
X(241) = X(1)-Hirst inverse of X(57)
X(241) = X(1)-line conjugate of X(55)
X(241) = X(I)-beth conjugate of X(J) for these (I,J): (2,241), (100,241), (1025,241), (1026,241)
X(242) = X(4)-LINE CONJUGATE OF X(71)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)[sin2A - sin B sin C]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(242) lies on these lines: 4,9 25,92 28,261 29,257 34,87 162,422 238,419 278,459 915,929
X(242) = isogonal conjugate of X(295)
X(242) = isotomic conjugate of X(337)
X(242) = X(4)-Hirst inverse of X(19)
X(242) = X(4)-line conjugate of X(71)
X(243) = X(4)-LINE CONJUGATE OF X(73)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)[cos2A - cos B cos C]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(243) lies on these lines: 1,4 3,158 55,92 65,412 318,958 411,821 425,662 522,652 920,1075 1040,1096
X(243) = isogonal conjugate of X(296)
X(243) = X(1)-Hirst inverse of X(4)
X(243) = X(1)-line conjugate of X(73)
X(244) = X(1)-LINE CONJUGATE OF X(100)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [1 - cos(B - C)]sin2(A/2)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(244) lies on these lines: 1,88 2,38 11,867 31,57 34,1106 42,354 58,229 63,748 238,896 474,976 518,899 596,1089 665,866
X(244) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,513), (75,514)
X(244) = crosspoint of X(1) and X(513)
X(244) = X(1)-line conjugate of X(100)
X(245) = X(1)-LINE CONJUGATE OF X(110)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = csc2(C - A) + csc(C - B) [csc(C - A) -csc(B - A)] + csc2(A - B)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(245) lies on these lines: 1,60 115,125
X(245) = X(1)-line conjugate of X(110)
X(246) = X(3)-LINE CONJUGATE OF X(110)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = csc(B-A)[cos A csc(B - A) + cos C csc(B - C)] - csc(C - A) u(A,B,C),
u(A,B,C) = [cos A csc(C - A) + cos B csc(C - B)]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(246) lies on these lines: 3,74 115,125
X(246) = X(3)-line conjugate of X(110)
X(247) = X(4)-LINE CONJUGATE OF X(110)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = csc(B-A)[sec A csc(B - A) + sec C csc(B - C)] - csc(C - A) u(A,B,C),
u(A,B,C) = [sec A csc(C - A) + sec B csc(C - B)]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(247) lies on these lines: 4,110 115,125
X(247) = X(4)-line conjugate of X(110)
X(248) = X(4)-LINE CONJUGATE OF X(132)
Trilinears sin 2A sec(A + ω) : sin 2B sec(B + ω) : sin 2C sec(C + ω)Barycentrics sin A sin 2A sec(A + ω) : sin B sin 2B sec(B + ω) : sin C sin 2C sec(C + ω)
X(248) lies on these lines:
4,32 6,157 39,54 50,67 65,172 66,571 69,287 72,293 74,187 290,385 682,695
X(248) = isogonal conjugate of X(297)
X(248) = crosspoint of X(98) and X(287)
X(248) = X(4)-line conjugate of X(132)
Centers 249- 297
are isogonal conjugates of previously listed centers.
X(249)
Trilinears (csc A)cos2(B - C) : (csc B)cos2(C - A) : (csc C)cos2(A - B)= a/(b2 - c2)2 : b/(c2 - a2)2 : c/(a2 - b2)2
Barycentrics cos2(B - C) : cos2(C - A) : cos2(A - B)
X(249) lies on these lines: 99,525 110,512 186,250 187,323 297,316 648,687 805,827 849,1110
X(249) = isogonal conjugate of X(115)
X(249) = isotomic conjugate of X(338)
X(249) = cevapoint of X(I) and X(J) for these (I,J): (6,110), (24,112)
X(249) = X(I)-cross conjugate of X(J) for these (I,J): (3,99), (6,110)
Let X = X(249) and let V be the vector-sum XA + XB + XC; then V = X(316)X(323).
X(250)
Trilinears (sec A)csc2(B - C) : (sec B)csc2(C - A) : (sec C)csc2(A - B)= (a2sec A)/(b2 - c2)2 : (b2sec B)/(c2 - a2)2 : (c2sec C)/(a2 - b2)2
Barycentrics (tan A)csc2(B - C) : (tan B)csc2(C - A) : (tan C)csc2(A - B)
X(250) lies on these lines: 23,232 107,687 110,520 112,691 186,249 325,340 476,933 523,648 827,935
X(250) = isogonal conjugate of X(125)
X(250) = isotomic conjugate of X(339)
X(250) = cevapoint of X(I) and X(J) for these (I,J): (3,110), (25,112), (162,270)
X(250) = X(I)-cross conjugate of X(J) for these (I,J): (3,110), (22,99), (24,107), (25,112), (199,101)
Let X = X(250) and let V be the vector-sum XA + XB + XC; then V = X(340)X(23).
X(251)
Trilinears a2csc(A + ω) : b2csc(B + ω) : c2csc(C + ω)= a/(b2 + c2) : b/(c2 + a2) : c/(a2 + b2)
Barycentrics a3csc(A + ω) : b3csc(B + ω) : c3csc(C + ω)
X(251) lies on these lines: 2,32 6,22 37,82 110,694 112,427 184,263 308,385 609,614 689,699
X(251) = isogonal conjugate of X(141)
X(251) = cevapoint of X(6) and X(32)
X(251) = X(I)-cross conjugate of X(J) for these (I,J): (6,83), (23,111), (523,112)
X(252)
Trilinears (cos A csc 2A)/f(A,B,C) : (cos B csc 2B)/f(B,C,A) : (cos C csc 2C)/f(C,A,B), wheref(A,B,C) = [(v + w)2][u4 + v4 + w4 - u2(2 v2 + 2w2 - vw) - vw(v2 + w2)], where
u = sin(2A), v = sin(2B), w = sin(2C).
Barycentrics 1/f(A,B,C) : 1/f(B,C,A) : 1/f(C,A,B)
X(252) lies on these lines: 3,930 54,140 93,186
X(252) = isogonal conjugate of X(143)
X(253) X(4)-CROSS CONJUGATE OF X(2)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)/(tan B + tan C - tan A)g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (csc2A)/(cos A - cos B cos C)
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/(tan B + tan C - tan A)
X(253) lies on these lines: 2,1073 7,280 8,307 20,64 193,287 306,329 318,342 322,341
X(253) = isogonal conjugate of X(154)
X(253) = isotomic conjugate of X(20)
X(253) = cyclocevian conjugate of X(69)
X(253) = cevapoint of X(122) and X(525)
X(253) = X(I)-cross conjugate of X(J) for these (I,J): (4,2), (122,525)
X(254) X(3)-CROSS CONJUGATE OF X(4)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)/(cos2B + cos2C - cos2A)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (tan A)/(cos2B + cos2C - cos2A)
X(254) lies on these lines: 2,847 4,155 24,393 46,225 68,136
X(254) = isogonal conjugate of X(155)
X(254) = cevapoint of X(136) and X(523)
X(254) = X(3)-cross conjugate of X(4)
X(255)
Trilinears cos2A : cos2B : cos2CBarycentrics sin A cos2A : sin B cos2B : sin C cos2C
X(255) lies on these lines: 1,21 3,73 35,991 36,1106 40,109 48,563 55,601 56,602 57,580 91,1109 92,1087 158,775 162,1099 165,1103 200,271 201,1060 219,268 293,304 326,1102 411,651 498,750 499,748
X(255) = isogonal conjugate of X(158)
X(255) = X(I)-Ceva conjugate of X(J) for these (I,J): (63,48), (283,3)
X(255) = crosspoint of X(63) and X(326)
X(255) = X(I)-aleph conjugate of X(J) for these (I,J): (775,255), (1105,158)
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(255); then W = X(225)X(255).
X(256)
Trilinears 1/(a2 + bc) : 1/(b2 + ca) : 1/(c2 + ab)Barycentrics a/(a2 + bc) : b/(b2 + ca) : c/(c2 + ab)
X(256) lies on these lines: 1,511 3,987 4,240 7,982 8,192 9,43 21,238 37,694 40,989 55,983 84,988 104,1064 291,894 314,350 573,981
X(256) = isogonal conjugate of X(171)
X(256) = X(239)-cross conjugate of X(291)
X(257)
Trilinears 1/(a3 + abc) : 1/(b3 + abc) : 1/(c3 + abc)Barycentrics a/(a3 + abc) : b/(b3 + abc) : c/(c3 + abc)
X(257) lies on these lines: 1,385 8,192 29,242 65,894 75,698 92,297 194,986 239,333 330,982 335,694
X(257) = isogonal conjugate of X(172)
X(527) = isotomic conjugate of X(894)
X(257) = X(350)-cross conjugate of X(335)
X(258) = CONGRUENT INCIRCLES ISOSCELIZER POINT
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/( cos B/2 + cos C/2 - cos A/2)= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1 + sin(B/2) + sin(C/2) - sin(A/2)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
In Yff's isoscelizer configuration, if X = X(258), then the isosceles triangles Ta, Tb, Tc have congruent incircles.
X(258) lies on these lines: 1,164 57,173 259,289
X(258) = isogonal conjugate of X(173)
X(258) = X(259)-cross conjugate of X(1)
X(258) = X(366)-aleph conjugate of X(363)
X(259)
Trilinears cos A/2 : cos B/2 : cos C/2= [a(b + c - a)]1/2 : [b(c + a - b)]1/2 : [c(a + b - c)]1/2
Barycentrics sin A cos A/2 : sin B cos B/2 : sin C cos C/2
X(259) lies on these lines: 1,168 258,289 260,266
X(259) = isogonal conjugate of X(174)
X(259) = X(I)-Ceva conjugate of X(J) for these (I,J): (174,266), (260,55)
X(259) = cevapoint of X(1) and X(503)
X(259) = crosspoint of X(I) and X(J) for these (I,J): (1,258), (174,188)
X(260)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A/2)/[cos B/2 + cos C/2 - cos A/2)]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(260) lies on these lines: 1,3 259,266
X(260) = isogonal conjugate of X(177)
X(260) = cevapoint of X(55) and X(259)
X(261)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [(csc A)(sec(B/2 - C/2))]2Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(261) lies on these lines:
2,593 9,645 21,314 28,242 58,86 75,99 272,310 284,332 317,406 319,502 552,873 572,662
X(261) = isogonal conjugate of X(181)
X(261) = isotomic conjugate of X(12)
X(261) = cevapoint of X(21) and X(333)
X(262)
Trilinears sec(A - ω) : sec(B - ω) : sec(C - ω)Barycentrics sin A sec(A - ω) : sin B sec(B - ω) : sin C sec(C - ω)
X(262) lies on these lines: 2,51 3,83 4,39 5,76 6,98 13,383 14,1080 25,275 30,598 226,982 381,671 385,576
X(262) = isogonal conjugate of X(182)
X(262) = isotomic conjugate of X(183)
Let X = X(262) and let V be the vector-sum XA + XB + XC; then V = X(76)X(4).
X(263)
Trilinears a2sec(A - ω) : b2sec(B - ω) : c2sec(C - ω)Barycentrics a3sec(A - ω) : b3sec(B - ω) : c3sec(C - ω)
X(263) lies on these lines: 2,51 6,160 69,308 184,251
X(263) = isogonal conjugate of X(183)
X(264) ISOTOMIC CONJUGATE OF CIRCUMCENTER
Trilinears csc A csc 2A : csc B csc 2B : csc C csc 2C= sec A csc2A : sec B csc2B : sec C csc2C
= tan A csc(A - ω) : tan B csc(B - ω) : tan C csc(C - ω)
Barycentrics csc 2A : csc 2B : csc 2C
X(264) lies on these lines:
2,216 3,95 4,69 5,1093 6,287 25,183 33,350 53,141 75,225 85,309 92,306 99,378 274,475 281,344 298,472 299,473 300,302 301,303 305,325 339,381 379,823 401,577
X(264) = isogonal conjugate of X(184)
X(264) = isotomic conjugate of X(3)
X(264) = anticomplement of X(216)
X(264) = X(276)-Ceva conjugate of X(2)
X(264) = cevapoint of X(I) and X(J) for these (I,J): (2,4), (5,324), (6,157), (92,318), (273,342), (338,523), (491,492)
X(264) = X(I)-cross conjugate of X(J) for these (I,J): (2,76), (5,2), (30,94), (92,331), (427,4), (442,321)
X(265)
Trilinears sin 2A csc 3A : sin 2B csc 3B : sin 2C csc 3C= 1/(4 cos A - sec A) : 1/(4 cos B - sec B) : 1/(4 cos C sec C)
Barycentrics sin A sin 2A csc 3A : sin B sin 2B csc 3B : sin C sin 2C csc 3C
X(265) lies on these lines: 3,125 4,94 5,49 6,13 30,74 64,382 65,79 67,511 69,328 290,316 300,621 301,622
X(265) = reflection of X(I) about X(J) for these (I,J): (3,125), (110,5), (399,113)
X(265) = isogonal conjugate of X(186)
X(265) = isotomic conjugate of X(340)
X(265) = cevapoint of X(5) and X(30)
X(265) = crosspoint of X(94) and X(328)
X(266)
Trilinears sin A/2 : sin B/2 : sin C/2= [bc(b + c - a)]1/2 : [ca(c + a - b)]1/2 : [ab(a + b - c)]1/2
Barycentrics sin A sin A/2 : sin B sin B/2 : sin C sin C/2
X(266) lies on these lines:1,164 56,289 174,188 259,260 361,978
X(266) = isogonal conjugate of X(188)
X(266) = eigencenter of cevian triangle of X(174)
X(266) = eigencenter of anticevian triangle of X(259)
X(266) = X(174)-Ceva conjugate of X(259)
X(266) = cevapoint of X(1) and X(361)
X(266) = X(6)-cross conjugate of X(289)
X(266) = crosspoint of X(1) and X(505)
X(267)
Trilinears f(a,b,c) : f(b,c,a) : f(ca,b), wheref(a,b,c) = 1/[b3 + c3 - a3 + (b + c - a)(bc + ca + ab)]
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(267) lies on these lines: 1,229 10,191 35,37
X(267) = reflection of X(1) about X(229)
X(267) = isogonal conjugate of X(191)
X(267) = cevapoint of X(58) and X(501)
X(267) = X(58)-cross conjugate of X(1)
X(268)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A cot A/2)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(268) lies on these lines: 3,9 21,280 219,255 220,577 222,1073 281,1012
X(268) = isogonal conjugate of X(196)
X(268) = X(I)-cross conjugate of X(J) for these (I,J): (48,219), (55,3)
X(269)
Trilinears tan2A/2 : tan2B/2 : tan2C/2Barycentrics sin A tan2A/2 : sin B tan2B/2 : sin C tan2C/2
X(269) lies on these lines: 1,7 3,939 6,57 9,241 46,1103 56,738 69,200 86,1088 106,934 142,948 273,1111 292,1020 307,936 320,326 479,614
X(269) = isogonal conjugate of X(200)
X(269) = isotomic conjugate of X(341)
X(269) = X(279)-Ceva conjugate of X(57)
X(269) = X(56)-cross conjugate of X(57)
X(269) = crosspoint of X(279) and X(479)
X(270)
Trilinears (sec A)/[1 + cos(B - C)] : (sec B)/[1 + cos(C - A)] : (sec C)/[1 + cos (A - B)]Barycentrics (tan A)/[1 + cos(B - C)] : (tan B)/[1 + cos(C - A)] : (tan C)/[1 + cos (A - B)]
X(270) lies on these lines: 4,162 27,58 28,60 29,283 759,933
X(270) = isogonal conjugate of X(201)
X(270) = X(250)-Ceva conjugate of X(162)
X(270) = cevapoint of X(28) and X(58)
X(270) = X(58)-cross conjugate of X(60)
X(271)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cot A cot A/2)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(271) lies on these lines: 2,1034 8,20 78,394 200,255 282,283
X(271) = isogonal conjugate of X(208)
X(271) = isotomic conjugate of X(342)
X(271) = X(I)-cross conjugate of X(J) for these (I,J): (3,78), (9,63)
X(272)
Trilinears f(A,B,C)/(b + c) : f(B,C,A)/(c + a) : f(C,A,B)/(a +b), wheref(A,B,C) = 1/[sin A + sin(A - B) + sin(A - C)]
Barycentrics af(A,B,C)/(b + c) : bf(B,C,A)/(c + a) : cf(C,A,B)/(a +b)
X(272) lies on these lines: 2,284 7,58 21,75 28,273 60,86 261,310 1014,1088
X(272) = isogonal conjugate of X(209)
X(272) = X(3)-cross conjugate of X(81)
X(273)
Trilinears sec A sec2(A/2) : sec B sec2(B/2) : sec C sec2(C/2)= (1- sec A)csc2A : (1 - sec B)csc2B : (1 - sec C)csc2C
Barycentrics tan A sec2(A/2) : tan B sec2(B/2) : tan C sec2(C/2)
X(273) lies on these lines: 2,92 4,7 19,653 27,57 28,272 29,34 53,1086 75,225 78,322 108,675 226,469 269,1111 317,320 458,894
X(273) = isogonal conjugate of X(212)
X(273) = isotomic conjugate of X(78)
X(273) = X(I)-Ceva conjugate of X(J) for these (I,J): (264,342), (286,7), (331,92)
X(273) = cevapoint of X(I) and X(J) for these (I,J): (4,278), (34,57)
X(273) = X(I)-cross conjugate of X(J) for these (I,J): (4,92), (57,85), (225,278)
X(274)
Trilinears b2c2/(b + c) : c2a2/(c + a) : a2b2/(a + b)= [a csc(A - ω)]/(b + c) : [b csc(B - ω)]/(c + a) :[c csc(C - ω)]/(a + b)
Barycentrics bc/(b + c) : ca/(c + a) : ab/(a + b)
X(274) lies on these lines:
1,75 2,39 7,959 10,291 21,99 28,242 57,85 58,870 69,443 81,239 88,799 110,767 183,474 213,894 264,475 278,331 315,377 325,442 961,1014
X(274) = isogonal conjugate of X(213)
X(274) = isotomic conjugate of X(37)
X(274) = X(310)-Ceva conjugate of X(314)
X(274) = cevapoint of X(I) and X(J) for these (I,J): (2,75), (85,348), (86,333)
X(274) = X(I)-cross conjugate of X(J) for these (I,J): (2,86), (75,310), (81,286), (333,314)
X(275) CEVAPOINT OF ORTHOCENTER AND SYMMEDIAN POINT
Trilinears csc 2A sec(B - C) : csc 2B sec(C - A) : csc 2C sec(A - B)Barycentrics sec A sec(B - C) : sec B sec(C - A) : sec C sec(A - B)
X(275) lies on these lines:
2,95 4,54 13,472 14,473 17,471 18,470 25,262 51,107 53,288 76,276 83,297 94,324 98,427
X(275) = isogonal conjugate of X(216)
X(275) = isotomic conjugate of X(343)
X(275) = X(276)-Ceva conjugate of X(95)
X(275) = cevapoint of X(4) and X(6)
X(275) = X(I)-cross conjugate of X(J) for these (I,J): (6,54), (54,95)
X(276)
Trilinears a3sec A sec(B - C) : b3sec B sec(C - A) : c3sec C sec(A - B)Barycentrics a4sec A sec(B - C) : b4sec B sec(C - A) : c4sec C sec(A - B)
X(276) lies on these lines: 3,95 4,327 54,290 76,275 97,401
X(276) = isogonal conjugate of X(217)
X(276) = isotomic conjugate of X(216)
X(276) = cevapoint of X(I) and X(J) for these (I,J): (2,264), (95,275)
X(276) = X(I)-cross conjugate of X(J) for these (I,J): (2,95), (401,290)
X(277)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = [sec2(A/2)]/[- cos4A/2 + cos4B/2 + cos4C/2]
Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
X(277) lies on these lines: 1,142 3,105 7,218 57,169 220,1086 241,278 942,1002
X(277) = isogonal conjugate of X(218)
X(277) = isotomic conjugate of X(345)
X(277) = X(55)-cross conjugate of X(7)
X(278)
Trilinears sec A tan A/2 : sec B tan B/2 : sec C tan C/2= csc A - 2 csc 2A : csc B - 2 csc 2B : csc C - 2 csc 2C
= (1 - sec A)/a : (1 - sec B)/b : (1 - sec C)/c
Barycentrics tan A tan A/2 : tan B tan B/2 : tan C tan C/2
= 1 - sec A : 1 - sec B : 1 - sec C
X(278) lies on these lines:
1,4 2,92 7,27 19,57 25,105 28,56 65,387 88,653 109,917 219,329 240,982 241,277 242,459 274,331 354,955 393,1108 412,962 443,1038 614,1096
X(278) = isogonal conjugate of X(219)
X(278) = isotomic conjugate of X(345)
X(278) = X(I)-Ceva conjugate of X(J) for these (I,J): (27,57), (92,196), (273,4), (331,7)
X(278) = cevapoint of X(19) and X(34)
X(278) = X(I)-cross conjugate of X(J) for these (I,J): (19,4), (56,7), (225,273)
X(279)
Trilinears csc A tan2A/2 : csc B tan2B/2 : csc C tan2C/2Barycentrics tan2A/2 : tan2B/2 : tan2C/2
X(279) lies on these lines: 1,7 2,85 28,1014 56,105 57,479 65,1002 144,220 145,664 304,346 942,955 985,1106
X(279) = isogonal conjugate of X(220)
X(279) = isotomic conjugate of X(346)
X(279) = cevapoint of X(57) and X(269)
X(279) = X(I)-cross conjugate of X(J) for these (I,J): (57,7), (269,479)
X(280) X(1)-CROSS CONJUGATE OF X(8)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc2A/2)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(280) lies on these lines: 2,318 7,253 8,20 21,268 75,309 78,282 285,1043 341,345
X(280) = isogonal conjugate of X(221)
X(280) = isotomic conjugate of X(347)
X(280) = X(309)-Ceva conjugate of X(189)
X(280) = cevapoint of X(1) and X(84)
X(280) = X(I)-cross conjugate of X(J) for these (I,J): (1,8), (281,2), (282,189)
X(281)
Trilinears sec A cot A/2 : sec B cot B/2 : sec C cot C/2= csc A + 2 csc 2A : csc B + 2 csc 2B : csc C + 2 csc 2C
= (1 + sec A)/a : (1 + sec B)/b : (1 + sec C)/c
Barycentrics tan A cot A/2 : tan B cot B/2 : tan C cot C/2
= 1 + sec A : 1 + sec B : 1 + sec C
X(281) lies on these lines:
1,282 2,92 4,9 7,653 8,29 28,958 33,200 37,158 45,53 48,944 100,1013 189,222 196,226 220,594 240,984 264,344 268,1012 318,346 380,950 451,1068 515,610 612,1096
X(281) = isogonal conjugate of X(222)
X(281) = isotomic conjugate of X(348)
X(281) = complement of X(347)
X(281) = X(I)-Ceva conjugate of X(J) for these (I,J): (29,33), (92,4)
X(281) = X(I)-cross conjugate of X(J) for these (I,J): (33,4), (37,9), (55,8)
X(281) = crosspoint of X(I) and X(J) for these (I,J): (2,280), (92,318)
X(282) X(6)-CROSS CONJUGATE OF X(9)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cot A/2)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(282) lies on these lines: 1,281 2,77 3,9 19,102 48,947 78,280 200,219 271,283 380,1036
X(282) = isogonal conjugate of X(223)
X(282) = X(189)-Ceva conjugate of X(84)
X(282) = X(I)-cross conjugate of X(J) for these (I,J): (6,9), (33,1)
X(282) = crosspoint of X(189) and X(280)
X(283)
Trilinears (cos A)/(cos B + cos C) : (cos B)/(cos C + cos A) : (cos C)/(cos A + cos B)Barycentrics (sin 2A)/(cos B + cos C) : (sin 2B)/(cos C + cos A) : (sin 2C)/(cos A + cos B)
X(283) lies on these lines: 1,21 2,580 3,49 29,270 60,284 77,603 78,212 86,307 102,110 271,282 474,582 643,1043 859,945 1010,1065
X(283) = isogonal conjugate of X(225)
X(283) = X(333)-Ceva conjugate of X(284)
X(283) = cevapoint of X(I) and X(J) for these (I,J): (3,255), (212,219)
X(283) = X(3)-cross conjugate of X(21)
X(283) = crosspoint of X(332) and X(333)
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(283); then W = X(407)X(283).
X(284)
Trilinears (sin A)/(cos B + cos C) : (sin B)/(cos C + cos A) : (sin C)/(cos A + cos B)Barycentrics a2/(cos B + cos C) : b2/(cos C + cos A) : c2/(cos A + cos B)
X(284) lies on these lines:
1,19 2,272 3,6 9,21 27,226 29,950 35,71 37,101 55,219 57,77 60,283 73,951 86,142 102,112 109,296 163,909 198,859 261,332 405,965 515,1065 942,1100
X(284) = isogonal conjugate of X(226)
X(284) = isotomic conjugate of X(349)
X(284) = inverse of X(579) in the Brocard circle
X(284) = X(I)-Ceva conjugate of X(J) for these (I,J): (81,58), (333,283)
X(284) = cevapoint of X(I) and X(J) for these (I,J): (6,48), (41,55)
X(284) = X(55)-cross conjugate of X(21)
X(284) = crosspoint of X(I) and X(J) for these (I,J): (21,81), (29,333)
X(285)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = 1/[(cos B + cos C)(-1 - cos A + cos B + cos C)]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(285) lies on these lines: 21,84 29,81 271,282 280,1043
X(285) = isogonal conjugate of X(227)
X(285) = X(58)-cross conjugate of X(21)
X(286)
Trilinears (csc 2A)/(sin B + sin C) : (csc 2B)/(sin C + sin A) : (csc 2C)/(sin A + sin B)Barycentrics (sec A)/(sin B + sin C) : (sec B)/(sin C + sin A) : (sec C)/(sin A + sin B)
X(286) lies on these lines: 4,69 7,331 19,27 28,242 29,34 99,915 112,767 158,969 322,1043
X(286) = isogonal conjugate of X(228)
X(286) = isotomic conjugate of X(72)
X(286) = cevapoint of X(I) and X(J) for these (I,J): (4,92), (7,273), (27,29), (28,81)
X(286) = X(I)-cross conjugate of X(J) for these (I,J): (4,27), (7,86), (81,274)
X(287)
Trilinears cot A sec(A + ω) : cot B sec(B + ω) : cot C sec(C + ω)Barycentrics cos A sec(A + ω) : cos B sec(B + ω) : cos C sec(C + ω)
X(287) lies on these lines:
2,98 6,264 69,248 83,217 95,141 185,384 193,253 293,306 297,685 305,394 401,511 651,894 879,895
X(287) = isogonal conjugate of X(232)
X(287) = isotomic conjugate of X(297)
X(287) = X(290)-Ceva conjugate of X(98)
X(287) = cevapoint of X(2) and X(401)
X(287) = X(248)-cross conjugate of X(98)
X(287) = X(2)-Hirst inverse of X(98)
X(288)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [sec(B - C)]/[b cos(C - A) + c cos(B - A)]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(288) lies on these lines: 51,54 53,275 97,216
X(288) = isogonal conjugate of X(233)
X(288) = cevapoint of X(6) and X(54)
X(289)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin A/2)/(cos B/2 + cos C/2 - cos A/2)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(289) lies on these lines: 1,363 56,266 258,259
X(289) = isogonal conjugate of X(236)
X(289) = X(6)-cross conjugate of X(266)
X(290)
Trilinears csc2A sec(A + ω) : csc2B sec(B + ω) : csc2C sec(C + ω)Barycentrics csc A sec(A + ω) : csc B sec(B + ω) : csc C sec(C + ω)
X(290) lies on these lines:
2,327 3,76 6,264 54,276 66,317 67,340 68,315 69,670 71,190 72,668 73,336 248,385 265,316 308,311 892,895
X(290) = isogonal conjugate of X(237)
X(290) = isotomic conjugate of X(511)
X(290) = cevapoint of X(I) and X(J) for these (I,J): (2,511), (98,287)
X(290) = X(I)-cross conjugate of X(J) for these (I,J): (385,308), (401,276), (511,2)
X(291)
Trilinears 1/(a2 - bc) : 1/(b2 - ca) : 1/(c2 - ab)Barycentrics a/(a2 - bc) : b/(b2 - ca) : c/(c2 - ab)
X(291) lies on these lines: 1,39 2,38 6,985 8,330 10,274 42,81 43,57 88,660 105,238 256,894 337,986 350,726 659,897 876,891
X(291) = isogonal conjugate of X(238)
X(291) = isotomic conjugate of X(350)
X(291) = X(I)-cross conjugate of X(J) for these (I,J): (239,256), (518,1)
X(291) = X(I)-Hirst inverse of X(J) for these (I,J): (1,292), (2,335)
X(292)
Trilinears a/(a2 - bc) : b/(b2 - ca) : c/(c2 - ab)Barycentrics a2/(a2 - bc) : b2/(b2 - ca) : c2/(c2 - ab)
X(292) lies on these lines: 1,39 2,334 6,869 9,87 37,86 44,660 58,101 106,813 171,893 269,1020 659,665
X(292) = isogonal conjugate of X(239)
X(292) = X(335)-Ceva conjugate of X(295)
X(292) = cevapoint of X(171) and X(238)
X(292) = X(1)-Hirst inverse of X(291)
X(293)
Trilinears cos A sec(A + ω) : cos B sec(B + ω) : cos C sec(C + ω)Barycentrics sin 2A sec(A + ω) : sin 2B sec(B + ω) : sin 2C sec(C + ω)
X(293) lies on these lines: 1,163 31,92 72,248 98,109 255,304 287,306
X(293) = isogonal conjugate of X(240)
X(294)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b2 + c2 - ab - ac)Barycentrics af(a,b,c) : bf(b,c,a) :cf(c,a,b)
X(294) lies on these lines: 1,41 2,949 4,218 6,7 8,220 19,1041 84,580 104,919 239,666 241,910 314,645
X(294) = isogonal conjugate of X(241)
X(294) = X(1)-Hirst inverse of X(105)
X(295)
Trilinears (cos A)/(a2 - bc) : (cos B)/(b2 - ca) : (cos C)/(c2 - ab)Barycentrics (sin 2A)/(a2 - bc) : (sin 2B)/(b2 - ca) : (sin 2C)/(c2 - ab)
X(295) lies on these lines: 27,335 43,57 58,101 72,337 103,813 150,334 875,926 876,928
X(295) = isogonal conjugate of X(242)
X(295) = X(335)-Ceva conjugate of X(292)
X(295) = crosspoint of X(335) and X(337)
X(296)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (cos A)/[cos2A - cos B cos C]
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = (sin 2A)/[cos2A - cos B cos C]
X(296) lies on these lines: 1,185 3,820 29,65 109,284
X(296) = isogonal conjugate of X(243)
X(297) X(2)-HIRST INVERSE OF X(4)
Trilinears csc 2A cos(A + ω) : csc 2B cos(B + ω) : csc 2C cos(C + ω)Barycentrics sec A cos(A + ω) : sec B cos(B + ω) : sec C cos(C + ω)
X(297) lies on these lines:
2,3 6,317 53,141 69,393 76,343 83,275 92,257 232,325 249,316 287,685 315,394 340,524 525,850
X(297) = isogonal conjugate of X(248)
X(297) = isotomic conjugate of X(287)
X(297) = inverse of X(458) in orthocentroidal circle
X(297) = complement of X(401)
X(297) = anticomplement of X(441)
X(297) = cevapoint of X(232) and X(511)
X(297) = X(511)-cross conjugate of X(325)
X(297) = X(2)-Hirst inverse of X(4)
Centers 298- 350
are isotomic conjugates of previously listed centers.
X(298) ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER
Trilinears csc2A sin(A + π/3) : csc2B sin(B + π/3) : csc2C sin(C + π/3)Barycentrics csc A sin(A + π/3) : csc B sin(B + π/3) : csc C sin(C + π/3)
X(298) lies on these lines:
2,6 3,617 5,634 13,532 14,76 15,533 18,636 99,531 140,628 264,472 316,530 317,473 319,1082 340,470 381,622 511,1080
X(298) = reflection of X(I) about X(J) for these (I,J): (299,325), (385,395)
X(298) = isotomic conjugate of X(13)
X(298) = anticomplement of X(396)
X(298) = X(300)-Ceva conjugate of X(303)
X(298) = X(15)-cross conjugate of X(470)
X(298) = X(2)-Hirst inverse of X(299)
X(299) ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER
Trilinears csc2A sin(A - π/3) : csc2B sin(B - π/3) : csc2C sin(C - π/3)Barycentrics csc A sin(A - π/3) : csc B sin(B - π/3) : csc C sin(C - π/3)
X(299) lies on these lines:
2,6 3,616 5,633 13,76 14,533 16,532 17,635 30,617 75,554 99,530 140,627 264,473 316,531 317,472 319,559 340,471 381,621 383,511
X(299) = reflection of X(I) about X(J) for these (I,J): (298,325), (385,396)
X(299) = isotomic conjugate of X(14)
X(299) = anticomplement of X(395)
X(299) = X(301)-Ceva conjugate of X(302)
X(299) = X(16)-cross conjugate of X(471)
X(299) = X(2)-Hirst inverse of X(298)
X(300) ISOTOMIC CONJUGATE OF 1st ISODYNAMIC CENTER
Trilinears csc2A csc(A + π/3) : csc2B csc(B + π/3) : csc2C csc(C + π/3)Barycentrics csc A sin(A + π/3) : csc B sin(B + π/3) : csc C sin(C + π/3)
X(300) lies on these lines: 2,94 13,76 264,302 265,621 303,311
X(300) = isotomic conjugate of X(15)
X(300) = cevapoint of X(298) and X(303)
X(300) = X(94)-Hirst inverse of X(301)
X(301) ISOTOMIC CONJUGATE OF 2nd ISODYNAMIC CENTER
Trilinears csc2A csc(A - π/3) : csc2B csc(B - π/3) : csc2C csc(C - π/3)Barycentrics csc A sin(A - π/3) : csc B sin(B - π/3) : csc C sin(C - π/3)
X(301) lies on these lines: 2,94 14,76 264,303 265,622 302,311
X(301) = isotomic conjugate of X(16)
X(301) = cevapoint of X(299) and X(302)
X(301) = X(94)-Hirst inverse of X(300)
X(302) ISOTOMIC CONJUGATE OF 1st NAPOLEON POINT
Trilinears csc2A csc(A + π/6) : csc2B csc(B + π/6) : csc2C csc(C + π/6)Barycentrics csc A sin(A + π/6) : csc B sin(B + π/6) : csc C sin(C + π/6)
X(302) lies on these lines:
2,6 3,621 5,622 14,99 16,316 18,76 61,629 140,633 264,300 301,311 317,470 381,616 549,617
X(302) = isotomic conjugate of X(17)
X(302) = X(301)-Ceva conjugate of X(299)
X(302) = X(61)-cross conjugate of X(473)
X(303) ISOTOMIC CONJUGATE OF 2nd NAPOLEON POINT
Trilinears csc2A csc(A - π/6) : csc2B csc(B - π/6) : csc2C csc(C - π/6)Barycentrics csc A sin(A - π/6) : csc B sin(B - π/6) : csc C sin(C - π/6)
X(303) lies on these lines:
2,6 3,622 5,621 13,99 15,316 17,76 62,630 140,634 264,301 300,311 317,471 381,617 549,616
X(303) = isotomic conjugate of X(18)
X(303) = X(300)-Ceva conjugate of X(298)
X(303) = X(62)-cross conjugate of X(472)
X(304)
Trilinears (cot A)csc2A : (cot B)csc2B : (cot C)csc2C= cos A csc(A - ω) : cos B csc(B - ω) : cos C csc(C - ω)
Barycentrics (cos A)csc2A : (cos B)csc2B : (cos C)csc2C
X(304) lies on these lines:
1,75 63,1102 69,72 76,85 92,561 255,293 279,346 305,306 309,322 341,1088 345,348
X(304) = isotomic conjugate of X(19)
X(304) = cevapoint of X(I) and X(J) for these (I,J): (63,326), (69,345), (312,322)
X(304) = X(I)-cross conjugate of X(J) for these (I,J): (63,75), (306,69)
X(305)
Trilinears b4c4cos A : c4a4cos B : a4b4cos C= cot A csc(A - ω) : cot B csc(B - ω) : cot C csc(C - ω)
Barycentrics b3c3cos A : c3a3cos B : a3b3cos C
X(305) lies on these lines:
2,39 22,99 25,683 95,183 264,325 287,394 304,306 311,1007 341,1088 350,614
X(305) = isotomic conjugate of X(25)
X(305) = X(69)-cross conjugate of X(76)
X(306)
Trilinears (b2c2)(b + c)cos A : (c2a2)(c + a)cos B : (a2b2)(a + b)cos CBarycentrics bc(b + c)cos A : ca(c + a)cos B : ab(a + b)cos C
X(306) lies on these lines:
1,2 27,1043 63,69 72,440 92,264 209,518 226,321 253,329 287,293 304,305 319,333
X(306) = isotomic conjugate of X(27)
X(306) = X(I)-Ceva conjugate of X(J) for these (I,J): (69, 72), (312,321), (313,10)
X(306) = X(I)-cross conjugate of X(J) for these (I,J): (71,10), (72,307), (440,2)
X(306) = crosspoint of X(I) and X(J) for these (I,J): (69,304), (312,345)
X(307)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2c2(b + c)(cos A)/(b + c - a)Barycentrics af(a,b,c) : bf(b,c,a) :cf(c,a,b)
X(307) lies on these lines: 2,7 8,253 69,73 75,225 86,283 95,320 141,241 269,936 319,664 948,966
X(307) = isotomic conjugate of X(29)
X(307) = X(349)-Ceva conjugate of X(226)
X(307) = X(I)-cross conjugate of X(J) for these (I,J): (72,306), (73,226)
X(307) = crosspoint of X(69) and X(75)
X(308)
Trilinears b3c3/(b2 + c2) : c3a3/(c2 + a2) : a3b3/(a2 + b2)= csc2A csc(A + ω) : csc2B csc(B + ω) : csc2C csc(C + ω)
= [csc(A - ω)]/(b2 + c2) : [csc(B - ω)]/(c2 + a2) : [csc(C - ω)]/(a2 + b2)
Barycentrics (b2c2)/(b2 + c2) : (c2a2)/(c2 + a2) : (a2b2)/(a2 + b2)
= csc A csc(A + ω) : csc B csc(B + ω) : csc C csc(C + ω)
X(308) lies on these lines: 2,702 6,76 25,183 42,313 69,263 111,689 141,670 251,385 290,311
X(308) = isotomic conjugate of X(39)
X(308) = cevapoint of X(2) and X(76)
X(308) = X(I)-cross conjugate of X(J) for these (I,J): (2,83), (385,290)
X(309)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc2A)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(309) lies on these lines: 69,189 75,280 77,318 84,314 85,264 304,322
X(309) = isotomic conjugate of X(40)
X(309) = cevapoint of X(189) and X(280)
X(309) = X(I)-cross conjugate of X(J) for these (I,J): (7,75), (92,85)
X(310)
Trilinears b3c3/(b + c) : c3a3/(c + a) : a3b3/(a + b)Barycentrics b2c2/(b + c) : c2a2/(c + a) : a2b2/(a + b)
X(310) lies on these lines: 2,39 7,314 38,75 86,350 99,675 261,272 321,335 333,673 670,903 871,982
X(310) = isotomic conjugate of X(42)
X(310) = cevapoint of X(I) and X(J) for these (I,J): (75,76), (274,314)
X(310) = X(75)-cross conjugate of X(274)
X(311)
Trilinears csc2A cos(B - C) : csc2B cos(C - A) : csc2C cos(A - B)Barycentrics csc A cos(B - C) : csc B cos(C - A) : csc C) cos(A - B)
X(311) lies on these lines: 2,570 4,69 22,157 53,324 95,99 141,338 290,308 300,303 301,302 305,1007
X(311) = isotomic conjugate of X(54)
X(311) = anticomplement of X(570)
X(311) = X(76)-Ceva conjugate of X(343)
X(311) = cevapoint of X(5) and X(343)
X(311) = X(5)-cross conjugate of X(324)
X(312)
Trilinears (b + c - a)b2c2 : (c + a - b)c2a2 : (a + b - c)a2b2= (1 + cos A)csc(A - ω) : (1 + cos B)csc(B - ω) : (1 + cos C)csc(C - ω)
Barycentrics bc(b + c - a) : ca(c + a - b) : ab(a + b - c)
X(312) lies on these lines: 1,1089 2,37 8,210 9,314 29,33 63,190 69,189 76,85 92,264 212,643 223,664 726,982 894,940 975,1010
X(312) = isogonal conjugate of X(604)
X(312) = isotomic conjugate of X(57)
X(312) = X(I)-Ceva conjugate of X(J) for these (I,J): (76,75), (304,322), (314,8)
X(312) = cevapoint of X(I) and X(J) for these (I,J): (2,329), (8,346), (9,78), (306,321)
X(312) = X(I)-cross conjugate of X(J) for these (I,J): (8,75), (9,318), (306,345), (346,341)
X(313)
Trilinears (b + c)b3c3 : (c + a)c3a3 : (a + b)a3b3= (b + c)csc(A - ω) : (c + a)csc(B - ω) : (a + b)csc(C - ω)
Barycentrics (b + c)b2c2 : (c + a)c2a2 : (a + b)a2b2
X(313) lies on these lines: 10,75 12,349 42,308 71,190 80,314 92,264 321,594 561,696
X(313) = isotomic conjugate of X(58)
X(313) = X(76)-Ceva conjugate of X(321)
X(313) = cevapoint of X(10) and X(306)
X(313) = X(321)-cross conjugate of X(349)
X(314)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2c2(b + c - a)/(b + c)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc(b + c - a)/(b + c)
X(314) lies on these lines:
1,75 2,941 4,69 6,9817,310 9,312 21,261 29,1039 58,987 79,320 80,313 81,321 84,309 99,104 256,350 294,645
X(314) = isotomic conjugate of X(65)
X(314) = X(310)-Ceva conjugate of X(274)
X(314) = cevapoint of X(I) and X(J) for these (I,J): (8,312), (69,75)
X(314) = X(I)-cross conjugate of X(J) for these (I,J): (8,333), (69,332), (333,274), (497,29)
X(315)
Trilinears bc(b4 + c4 - a4) : ca(c4 + a4 - b4) : ab(a4 + b4 - c4)Barycentrics b4 + c4 - a4 : c4 + a4 - b4 : a4 + b4 - c4
X(315) lies on these lines:
2,32 3,325 4,69 5,183 8,760 20,99 68,290 192,746 194,736 274,377 297,394 343,458 371,491 372,492 631,1007
X(315) = isotomic conjugate of X(66)
X(315) = anticomplement of X(32)
X(315) = X(I)-cross conjugate of X(J) for these (I,J): (206,2)
X(316) = DROUSSENT PIVOT
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b4 + c4 - a4 - b2c2)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b4 + c4 - a4 - b2c2
The reflection of X(99) in the polar of X(&6).
Lucien Droussent, "Cubiques circularies anallagmatiques par points réciproques ou isogonaux," Mathesis 62 (1953) 204-215.
X(316) lies on these lines:
2,187 4,69 15,303 16,302 30,99 115,385 148,538 183,381 249,297 265,290 298,530 299,531 376,1007 384,626 512,850 524,671 691,858
X(316) = midpoint between X(621) and X(622)
X(316) = reflection of X(I) about X(J) for these (I,J): (99,325), (385,115)
X(316) = isotomic conjugate of X(67)
X(316) = anticomplement of X(187)
X(317)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A cos 2A csc2ABarycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = tan A cos 2A csc2A
X(317) lies on these lines:
2,95 4,69 6,297 25,325 53,524 66,290 141,458 183,427 193,393 261,406 273,320 298,473 299,472 302,470 303,471 318,319 459,1007
X(317) = isotomic conjugate of X(68)
X(317) = anticomplement of X(577)
X(317) = cevapoint of X(52) and X(467)
X(318)
Trilinears (1 + sec A)/a2 : (1 + sec B)/b2 : (1 + sec C)/c2= sec A csc2A/2 : sec B csc2B/2 : sec C csc2C/2
Barycentrics (1 + sec A)/a : (1 + sec B)/b : (1 + sec C)/c
X(318) lies on these lines:
2,280 4,8 10,158 29,33 53,594 63,412 75,225 77,309 108,404 200,1089 208,653 239,458 243,958 253,342 281,346 317,319 475,1068
X(318) = isogonal conjugate of X(603)
X(318) = isotomic conjugate of X(77)
X(318) = X(264)-Ceva conjugate of X(92)
X(318) = cevapoint of X(9) and X(33)
X(318) = X(I)-cross conjugate of X(J) for these (I,J): (9,312), (10,8), (281,92)
X(319)
Trilinears (1 + 2 cos A)/a2 : (1 + 2 cos B)/b2 : (1 + 2 cos C)/c2Barycentrics (1 + 2 cos A)/a : (1 + 2 cos B)/b : (1 + 2 cos C)/c
X(319) lies on these lines: 2,1100 7,8 10,86 80,313 141,239 171,757 200,326 261,502 298,1082 299,559 306,333 307,664 317,318 321,1029 344,391 524,594
X(319) = isotomic conjugate of X(79)
X(319) = anticomplement of X(1100)
X(320)
Trilinears (1 - 2 cos A)/a2 : (1 - 2 cos B)/b2 : (1 - 2 cos C)/c2Barycentrics (1 - 2 cos A)/a : (1 - 2 cos B)/b : (1 - 2 cos C)/c
X(320) lies on these lines:
1,752 2,44 7,8 58,86 79,314 95,307 141,894 144,344 190,527 239,524 269,326 273,317 334,660 350,513 519,679
X(320) = isotomic conjugate of X(80)
X(320) = X(214)-cross conjugate of X(1)
X(321)
Trilinears (b + c)b2c2 : (c + a)c2a2 : (a + b)a2b2= a(b + c)csc(A - ω) : b(c + a)csc(B - ω) : c(a + b)csc(C - ω)
Barycentrics bc(b + c) : ca(c + a) : ab(a + b)
X(321) lies on these lines:
1,964 2,37 4,8 10,756 38,726 76,561 81,314 83,213 98,100 190,333 226,306 310,335 313,594 319,1029 668,671 693,824
X(321) = isotomic conjugate of X(81)
X(321) = X(I)-Ceva conjugate of X(J) for (I,J) = (75,10), (76,313), (312,306)
X(321) = cevapoint of X(37) and X(72)
X(321) = X(442)-cross conjugate of X(264)
X(321) = crosspoint of X(I) and X(J) for these (I,J): (75,76), (313,349)
X(322)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (-1 - cos A + cos B + cos C)csc2ABarycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (-1 - cos A + cos B + cos C)csc A
X(322) lies on these lines: 2,1108 7,8 78,273 92,264 227,347 253,341 286,1043 304,309 326,664
X(322) = isotomic conjugate of X(84)
X(322) = anticomplement of X(1108)
X(322) = X(304)-Ceva conjugate of X(312)
X(322) = X(347)-cross conjugate of X(75)
X(323)
Trilinears sin 3A csc2A : sin 3B csc2B : sin 3C csc2CBarycentrics sin 3A csc A : sin 3B csc B : sin 3C csc C
X(323) lies on these lines: 2,6 20,155 23,110 30,146 140,195 187,249 401,525
X(323) = reflection of X(23) about X(110)
X(323) = isotomic conjugate of X(94)
X(323) = X(340)-Ceva conjugate of X(186)
X(323) = cevapoint of X(6) and X(399)
X(323) = X(50)-cross conjugate of X(186)
X(324)
Trilinears bc sec A cos(B - C) : ca sec B cos(C - A) : ab sec C cos(A - B)Barycentrics sec A cos(B - C) : sec B cos(C - A) : sec C cos(A - B)
X(324) lies on these lines: 2,216 4,52 53,311 94,275 110,436 143,565
X(324) = isotomic conjugate of X(97)
X(324) = X(264)-Ceva conjugate of X(5)
X(324) = cevapoint of X(I) and X(J) for these (I,J): (5,53), (52,216)
X(324) = X(5)-cross conjugate of X(311)
X(325)
Trilinears csc2A cos(A + ω) : csc2B cos(B + ω) : csc2C cos(C + ω)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b4 + c4 - a2b2 - a2c2)
Barycentrics g(a,b,c) : g(b,c,a) : b(c,a,b), where g(a,b,c) = b4 + c4 - a2b2 - a2c2
X(325) lies on these lines:
2,6 3,315 5,76 11,350 22,160 25,317 30,99 39,626 114,511 115,538 187,620 232,297 250,340 264,305 274,442 383,622 523,684 621,1080
X(325) = midpoint between X(I) and X(J) for these (I,J): (99,316), (298,299)
X(325) = reflection of X(385) about X(230)
X(325) = complement of X(385)
X(325) = anticomplement of X(230)
X(325) = cevapoint of X(2) and X(147)
X(325) = X(I)-cross conjugate of X(J) for these (I,J): (114,2), (511,297)
X(325) = X(2)-Hirst inverse of X(69)
X(326)
Trilinears cot2A : cot2B : cot2CBarycentrics csc A - sin A : csc B - sin B : csc C - sin C
X(326) lies on these lines: 1,75 48,63 69,73 200,319 255,1102 269,320 322,664 610,662
X(326) = isogonal conjugate of X(1096)
X(326) = isotomic conjugate of X(158)
X(326) = X(I)-Ceva conjugate of X(J) for these (I,J): (304,63), (332,69)
X(326) = X(255)-cross conjugate of X(63)
X(327)
Trilinears csc2A sec(A - ω) : csc2B sec(B - ω) : csc2C sec(C - ω)= sin A csc(2A - 2 ω): sin B csc(2B - 2 ω) : sin C csc(2C - 2 ω)
Barycentrics csc A sec(A - ω) : csc B sec(B - ω) : csc C sec(C - ω)
X(327) lies on these lines: 2,290 4,276 5,76 53,141 69,263 95,160
X(327) = isotomic conjugate of X(182)
X(328)
Trilinears cot A csc 3A : cot B csc 3B : cot C csc 3CBarycentrics cos A csc 3A : cos B csc 3B : cos C csc 3C
X(328) lies on these lines: 2,94 69,265 95,99
X(328) = isotomic conjugate of X(186)
X(328) = X(265)-cross conjugate of X(94)
X(329)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (-1 - cos A + cos B + cos C)(csc A)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = -1 - cos A + cos B + cos C
X(329) lies on these lines:
1,452 2,7 4,8 20,78 55,1005 69,189 100,972 190,345 191,498 196,342 200,516 219,278 220,948 223,347 253,306 388,960 392,1056 394,651 405,999 497,518
X(329) = isotomic conjugate of X(189)
X(329) = cyclocevian conjugate of X(1034)
X(329) = anticomplement of X(57)
X(329) = X(I)-Ceva conjugate of X(J) for (I,J) = (69,8), (312,2)
X(329) = X(I)-cross conjugate of X(J) for these (I,J): (40,347), (223,2)
X(330)
Trilinears bc/(ab + ac - bc) : ca/(bc + ba - ca) : ab/(ca + cb - ab)Barycentrics 1/(ab + ac - bc) : 1/(bc + ba - ca) : 1/(ca + cb - ab)
X(330) lies on these lines: 1,87 2,1107 8,291 56,385 57,239 76,1015 105,932 145,1002 193,959 257,982
X(330) = isotomic conjugate of X(192)
X(330) = X(87)-Ceva conjugate of X(2)
X(330) = X(75)-cross conjugate of X(2)
X(331)
Trilinears sec2A csc(2A) : sec2B csc(2B) : sec2C csc(2C)= (1 - sec A)csc(A - ω) : (1 - sec B)csc(B - ω) : (1 - sec C)csc(C - ω)
Barycentrics sec3A : sec3B : sec3C
X(331) lies on these lines: 4,150 7,286 34,870 75,225 85,92 108,767 274,278
X(331) = isotomic conjugate of X(219)
X(331) = cevapoint of X(I) and X(J) for these (I,J): (7,278), (92,273)
X(331) = X(92)-cross conjugate of X(264)
X(332)
Trilinears (cot A csc A)/(cos B + cos C) : (cot B csc B)/(cos C + cos A) : (cot C csc C)/(cos A + cos B)Barycentrics (cot A)/(cos B + cos C) : (cot B)/(cos C + cos A) : (cot C)/(cos A + cos B)
X(332) lies on these lines: 1,75 3,69 21,1036 99,102 219,345 261,284 1014,1037
X(332) = isotomic conjugate of X(225)
X(332) = cevapoint of X(I) and X(J) for these (I,J): (69,326), (78,345)
X(332) = X(I)-cross conjugate of X(J) for these (I,J): (69,314), (283,333)
X(333) CEVAPOINT OF X(8) AND X(9)
Trilinears bc(b + c - a)/(b + c) : ca(c + a - b)/(c + a) : ab(a + b - c)/(a + b)Barycentrics (b + c - a)/(b + c) : (c + a - b)/(c + a) : (a + b - c)/(a + b)
X(333) lies on these lines:
2,6 8,21 9,312 10,58 19,27 29,270 57,85 190,321 239,257 261,284 306,319 310,673 662,909 740,846 859,956 1021,1024
X(333) = isotomic conjugate of X(226)
X(333) = X(I)-Ceva conjugate of X(J) for these (I,J): (261,21), (274,86)
X(333) = cevapoint of X(I) and X(J) for these (I,J): (2,63), (8,9), (283,284)
X(333) = X(I)-cross conjugate of X(J) for these (I,J): (8,314), (9,21), (21,86), (283,332), (284,29)
X(333) = crosspoint of X(274) and X(314)
X(334)
Trilinears b2c2/(a2 - bc) : c2a2/(b2 - ca) : a2b2/(c2 - ab)Barycentrics bc/(a2 - bc) : ca/(b2 - ca) : ab/(c2 - ab)
X(334) lies on these lines: 2,292 10,274 12,85 75,141 76,1089 150,295 320,660 741,839 767,813
X(334) = isotomic conjugate of X(238)
X(334) = X(75)-Hirst inverse of X(335)
X(335)
Trilinears bc/(a2 - bc) : ca/(b2 - ca) : ab/(c2 - ab)Barycentrics 1/(a2 - bc) : 1/(b2 - ca) : 1/(c2 - ab)
X(335) lies on these lines: 1,384 2,38 7,192 27,295 37,86 75,141 76,871 239,518 257,694 310,321 320,742 536,903 675,813 741,835 876,900
X(335) = reflection of X(190) about X(37)
X(335) = isotomic conjugate of X(239)
X(335) = cevapoint of X(I) and X(J) for these (I,J): (37,518), (292,295)
X(335) = X(I)-cross conjugate of X(J) for these (I,J): (295,337), (350,257)
X(335) = X(I)-Hirst inverse of X(J) for these (I,J): (2,291), (75,334)
X(336)
Trilinears csc A cot A sec(A + ω) : csc B cot B sec(B + ω) : csc C cot C sec(C + ω)Barycentrics cot A sec(A + ω) : cot B sec(B + ω) : cot C sec(C + ω)
X(336) lies on these lines: 1,811 48,75 73,290 255,293
X(336) = isotomic conjugate of X(240)
X(337)
Trilinears (csc A cot A)/(a2 - bc) : (csc B cot B))/(b2 - ca) : (csc C cot C)/(c2 - ab)Barycentrics (cot A)/(a2 - bc) : (cot B)/(b2 - ca) : (cot C)/(c2 - ab)
X(337) lies on these lines: 12,85 37,86 72,295 201,348 291,986
X(337) = isotomic conjugate of X(242)
X(337) = X(295)-cross conjugate of X(335)
X(338) CEVAPOINT OF X(115) AND X(125)
Trilinears (b2 - c2)2/a3 : (c2 - a2)2/b3 : (a2 - b2)2/c3= csc A sin2(B - C) : csc B sin2(C - A) : csc C sin2(A - B)
Barycentrics (b2 - c2)2/a2 : (c2 - a2)2/b2 : (a2 - b2)2/c2
= csc A sin2(B - C) : csc B sin2(C - A) : csc C sin2(A - B)
X(338) lies on these lines:
2,94 4,67 6,264 50,401 76,599 115,127 125,136 141,311
X(338) = isotomic conjugate of X(249)
X(338) = X(264)-Ceva conjugate of X(523)
X(338) = cevapoint of X(115) and X(125)
X(338) = X(125)-cross conjugate of X(339)
X(339)
Trilinears (b2 - c2)2(cos A)/a4 : (c2 - a2)2(cos B)/b4 : (a2 - b2)2(cos C)/c4= csc A cot A sin2(B - C) : csc B cot B sin2(C - A) : csc C cot C sin2(A - B)
Barycentrics (b2 - c2)2(cos A)/a3 : (c2 - a2)2(cos B)/b3 : (a2 - b2)2(cos C)/c3
= cot A sin2(B - C) : cot B sin2(C - A) : cot C sin2(A - B)
X(339) lies on these lines: 3,76 69,265 115,127 264,381
X(339) = isotomic conjugate of X(250)
X(339) = X(76)-Ceva conjugate of X(525)
X(339) = X(125)-cross conjugate of X(338)
X(340)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = sec A sin 3A csc3A : sec B sin 3B csc3B : sec C sin 3C csc3C
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = sec A sin 3A csc2A : sec B sin 3B csc2B : sec C sin 3C csc2C
X(340) lies on these lines: 4,69 67,290 95,140 250,325 297,524 298,470 299,471 447,540 458,599 520,850
X(340) = isotomic conjugate of X(265)
X(340) = cevapoint of X(186) and X(323)
X(341)
Trilinears csc4A/2 : csc4B/2 : csc4C/2Barycentrics sin A csc4A/2 : sin B csc4B/2 : sin C csc4C/2
X(341) lies on these lines: 1,1050 8,210 10,75 40,190 200,1043 253,322 280,345 304,668 305,1088
X(341) = isogonal conjugate of X(1106)
X(341) = isotomic conjugate of X(269)
X(341) = X(346)-cross conjugate of X(312)
X(342)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc 2A tan A/2)/(-1 - cos A + cos B + cos C)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sec A tan A/2)/(-1 - cos A + cos B + cos C)
X(342) lies on these lines: 4,7 9,653 85,264 92,226 108,1005 196,329 253,318 393,948
X(342) = isotomic conjugate of X(271)
X(342) = X(I)-Ceva conjugate of X(J) for these (I,J): (85,92), (264,273)
X(342) = cevapoint of X(208) and X(223)
X(343)
Trilinears cot A cos(B - C) : cot B cos(C - A) : cot C cos(A - B)Barycentrics cos A cos(B - C) : cos B cos(C - A) : cos C cos(A - B)
X(343) lies on these lines:
2,6 3,68 5,51 22,161 53,311 76,297 140,569 315,458 427,511 470,634 471,633 472,621 473,622
X(343) = isotomic conjugate of X(275)
X(343) = X(I)-Ceva conjugate of X(J) for these (I,J): (76,311), (311,5)
X(343) = X(216)-cross conjugate of X(5)
X(343) = crosspoint of X(69) and X(76)
X(344)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (csc2A/2)[cos4(B/2) + cos4(C/2) - cos4(A/2)]
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)
X(344) lies on these lines:
2,37 7,190 8,480 9,69 44,193 45,141 144,320 264,281 319,391
X(344) = isotomic conjugate of X(277)
X(345)
Trilinears (csc A)/(1 - sec A) : (csc B)/(1 - sec B) : (csc C)/(1 - sec C)Barycentrics 1/(1 - sec A) : 1/(1 - sec B) : 1/(1 - sec C)
X(345) lies on these lines:
2,37 8,21 22,100 57,728 63,69 78,1040 190,329 219,332 280,341 304,348 498,1089
X(345) = isogonal conjugate of X(608)
X(345) = isotomic conjugate of X(278)
X(345) = X(I)-Ceva conjugate of X(J) for these (I,J): (304,69), (332,78)
X(345) = X(I)-cross conjugate of X(J) for these (I,J): (78,69), (219,8), (306,312)
X(346)
Trilinears bc(b + c - a)2 : ca(c + a - b)2 : ab(a + b - c)2= cos(A/2) csc3(A/2) : cos(B/2) csc3(B/2) : cos(C/2) csc3(C/2)
Barycentrics (b + c - a)2 : (c + a - b)2 : (a + b - c)2
X(346) lies on these lines:
2,37 6,145 8,9 45,594 69,144 78,280 100,198 219,644 220,1043 253,306 279,304 281,318 573,1018
X(346) = isotomic conjugate of X(279)
X(346) = X(312)-Ceva conjugate of X(8)
X(346) = X(200)-cross conjugate of X(8)
X(346) = crosspoint of X(312) and X(341)
X(347)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (-1 - cos A + cos B + cos C) sec2(A/2)Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
X(347) lies on these lines:
1,7 2,92 8,253 34,452 37,948 69,664 75,280 144,219 223,329 227,322 241,1108 573,1020
X(347) = isotomic conjugate of X(280)
X(347) = anticomplement of X(281)
X(347) = X(I)-Ceva conjugate of X(J) for these (I,J): (75,7), (348,2)
X(347) = cevapoint of X(40) and X(223)
X(347) = X(I)-cross conjugate of X(J) for these (I,J): (40,329), (221,196), (227,223)
X(347) = crosspoint of X(75) and X(322)
X(348)
Trilinears cot A sec2(A/2) : cot B sec2(B/2) : cot C sec2(C/2)= (csc A)/(1 + sec A) : (csc B)/(1 + sec B) : (csc C)/(1 + sec C)
Barycentrics 1/(1 + sec A) : 1/(1 + sec B) : 1/(1 + sec C)
X(348) lies on these lines: 2,85 7,21 8,664 69,73 75,280 150,944 201,337 274,278 304,345 499,1111
X(348) = isogonal conjugate of X(607)
X(348) = isotomic conjugate of X(281)
X(348) = X(274)-Ceva conjugate of X(85)
X(348) = cevapoint of X(I) and X(J) for these (I,J): (2,347), (63,77)
X(348) = X(222)-cross conjugate of X(7)
X(349)
Trilinears (cos B + cos C)csc3A : (cos C + cos A)csc3B : (cos A + cos B) csc3C= (cos B + cos C)csc(A - ω) : (cos C + cos A)csc(B - ω) : (cos A + cos B)csc(C - ω)
Barycentrics (cos B + cos C)csc2A : (cos C + cos A)csc2B : (cos A + cos B)(csc C/2)2
X(349) lies on these lines: 12,313 73,290 75,225 76,85
X(349) = isotomic conjugate of X(284)
X(349) = cevapoint of X(226) and X(307)
X(349) = X(321)-cross conjugate of X(313)
X(350)
Trilinears (a2 - bc)b2c2 : (b2 - ca)c2a2 : (c2 - ab)a2b2Barycentrics bc(a2 - bc) : ca(b2 - ca) : ab(c2 - ab)
X(350) lies on these lines:
1,76 2,37 11,325 33,264 36,99 42,308 55,183 69,497 86,310 172,384 190,672 256,314 291,726 305,614 320,513 447,811 519,668 538,1015 889,903
X(350) = isotomic conjugate of X(291)
X(350) = crosspoint of X(257) and X(335)
X(350) = X(2)-Hirst inverse of X(75)
X(351) = CENTER OF THE PARRY CIRCLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)(b2 + c2 - 2a2)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 - c2)(b2 + c2 - 2a2)
X(351) is the center of the Parry circle introduced in TCCT (Art. 8.13) as the circle that passes through X(i) for i = 2, 15, 16, 23, 110, 111, 352, 353.
X(351) lies on these lines: 2,804 110,526 184,686 187,237 694,881 865,888
X(351) = isogonal conjugate of X(892)
X(351) = crosspoint of X(110) and X(111)
X(352)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(-a4 - b4 - c4 - 5b2c2 + 4a2b2 + 4a2c2)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
A point on the Parry circle; see X(351).
X(352) lies on these lines: 2,6 3,353 110,187 111,511
X(352) = inverse of X(353) in the circumcircle
X(353)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(4a4 - 2b4 - 2c4 - b2c2 - 4a2b2 - 4a2c2)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
A point on the Parry circle; see X(351).
X(353) lies on these lines: 3,352 6,23 110,574 111,182
X(353) = inverse of X(352) in the circumcircle
X(353) = inverse of X(111) in the Brocard circle
X(354) = WEILL POINT
Trilinears (b - c)2 - ab - ac : (c - a)2 - bc - ba : (a - b)2 - ca - cbBarycentrics a[(b - c)2 - ab - ac] : b[(c - a)2 - bc - ba] : c[(a - b)2 - ca - cb]
X(354) is the centroid of the intouch triangle.
William Gallatly, The Modern Geometry of the Triangle, 2nd edition, Hodgson, London, 1913, page 16.
X(354) lies on these lines: 1,3 2,210 6,374 7,479 11,118 37,38 42,244 44,748 48,584 63,1001 81,105 278,955 373,375 388,938 392,551 516,553
X(354) = reflection of X(I) about X(J) for these (I,J): (210,2)
X(354) = X(101)-Ceva conjugate of X(513)
X(354) = crosspoint of X(1) and X(7)
Let X = X(354) and let V be the vector-sum XA + XB + XC; then V = X(72)X(1) = X(8)X(65).
X(355) = FUHRMANN CENTER
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a cos A - (b + c)cos(B - C)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
The center of the Fuhrmann circle, defined as the circumcircle of the Furhmann triangle A"B"C", where A" is obtained as follows: let A' be the midpoint of the shorter arc having endpoints B and C on the circumcircle of ABC; then A" is the reflection of A' about line BC. Vertices B" and C" are obtained cyclically.
Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 6: The Fuhrmann Circle.
X(355) lies on these lines:
1,5 2,944 3,10 4,8 30,40 65,68 85,150 104,404 165,550 381,519 382,516 388,942 938,1056
X(355) = midpoint between X(4) and X(8)
X(355) = reflection of X(I) about X(J) for these (I,J): (1,5), (3,10)
X(355) = complement of X(944)
X(356) = 1st MORLEY CENTER
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A/3 + 2 cos B/3 cos C/3Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(356) is the centroid of the Morley equilateral triangle. Triangle centers bearing Morley's name possibly do not appear in the pre-1994 literature on Morley's famous theorem. For a discussion of the theorem and extensive list of references, see
C. O. Oakley and J. C. Baker, "The Morley trisector theorem," American Mathematical Monthly 85 (1978) 737-745.
X(356) lies on this line: 357,358
X(357) = 2nd MORLEY CENTER
Trilinears sec A/3 : sec B/3 : sec C/3Barycentrics sin A sec A/3 : sin B sec B/3 : sin C sec C/3
X(357) is the perspector of Morley triangle and ABC.
X(357) lies on this line: 356,358
X(357) = isogonal conjugate of X(358)
X(358) = MORLEY-YFF CENTER
Trilinears cos A/3 : cos B/3 : cos C/3Barycentrics sin A cos A/3 : sin B cos B/3 : sin C cos C/3
X(358) is the perspector of the adjunct Morley triangle and ABC.
X(358) lies on this line: 356,357
X(358) = isogonal conjugate of X(357)
X(359) = HOFSTADTER ONE POINT
Trilinears a/A : b/B : c/CBarycentrics 1/A : 1/B : 1/C
This point is the limit as r approaches 1 of the perspector of the r-Hofstadter triangle and ABC. See X(360) for details.
X(359) = isogonal conjugate of X(360)
X(360) = HOFSTADTER ZERO POINT
Trilinears A/a : B/b : C/cBarycentrics A : B : C
This point is obtained as a limit of perspectors. Let r denote a real number, but not 0 or 1. Using vertex B as a pivot, swing line BC toward vertex A through angle rB and swing line BC about C through angle rC. Let A(r) be the point in which the two swung lines meet. Obtain B(r) and C(r) cyclically. Triangle A(r)B(r)C(r) is the r-Hofstadter triangle; its perspector with ABC is the point given by trilinears
sin(r(A))/sin(A - r(A)) : sin(r(B))/sin(B - r(B)) : sin(r(C))/sin(C - r(C)).
The limit of this point as r approaches 0 is X(360). The two Hofstadter points, X(359) and X(360) are examples of transcendental triangle centers, since they have no trilinear or barycentric representation using only algebraic functions of a,b,c (or sin A, sin B, sin C).
Clark Kimberling, "Hofstadter points," Nieuw Archief voor Wiskunde 12 (1994) 109-114.
X(360) = isogonal conjugate of X(359)
X(361)
Trilinears csc B/2 + csc C/2 - csc A/2 : csc C/2 + csc A/2 - csc B/2 : csc A/2 + csc B/2 - csc C/2Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin A)(csc B/2 + csc C/2 - csc A/2)
The isoscelizer equation au(X) = bv(X) = cw(X) has solution X = X(361).
X(361) lies on these lines: 1,188 164,503 266,978
X(361) = X(266)-Ceva conjugate of X(1)
X(362) = CONGRUENT CIRCUMCIRCLES ISOSCELIZER POINT
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = b cos B/2 + c cos C/2 - a cos A/2
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
The isoscelizer equations u(X)/a = v(X)/b = w(X)/c have solution X = X(362).
X(362) lies on this line: 57,234
X(362) = X(508)-Ceva conjugate of X(1)
X(363) = EQUAL PERIMETERS ISOSCELIZER POINT
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b/(1 + sin B/2) + c/(1 + sin C/2) - a/(1 + sin A/2)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
When X = X(363), the isoscelizer triangles have equal perimeters.
X(363) lies on these lines: 1,289 40,164 165,166
X(364) = WABASH CENTER (EQUAL AREAS ISOSCELIZER POINT)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b1/2 + c1/2 - a1/2Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
When X = X(364), the isoscelizer triangles T(X,a), T(X,b), T(X,c) have equal areas.
X(364) lies on these lines: 1,365 9,366
X(364) = X(366)-Ceva conjugate of X(1)
X(365) = SQUARE ROOT POINT
Trilinears a1/2 : b1/2 : c1/2Barycentrics a3/2 : b3/2 : c3/2
For a construction of X(365), see the note at X(2), which provides for a construction barycentric square roots which one can easily extend to a construction for trilinear square roots.
X(365) lies on this line: 1,364
X(365) = isogonal conjugate of X(366)
X(366)
Trilinears a-1/2 : b-1/2 : c-1/2Barycentrics a1/2 : b1/2 : c1/2
See the note at X(365).
X(366) lies on these lines: 2,367 9,364
X(366) = isogonal conjugate of X(365)
X(366) = cevapoint of X(1) and X(364)
X(366) = X(367)-cross conjugate of X(1)
X(367)
Trilinears b1/2 + c1/2 : c1/2 + a1/2 : a1/2 + b1/2Barycentrics a1/2(b1/2 + c1/2) : b1/2(c1/2 + a1/2) : c1/2(a1/2 + b1/2)
X(367) lies on these lines: 1,364 2,366
X(367) = crosspoint of X(1) and X(366)
X(368) = EQUI-BROCARD CENTER
Trilinears (reasonable trilinears are sought)Barycentrics (reasonable barycentrics are sought)
The center X for which the triangle XBC, XCA, XAB have equal Brocard angles. Yff proved that X(368) lies on the self-isogonal conjugate cubic with trilinear equation f(a,b,c)u + f(b,c,a)v + f(c,a,b)w = 0, where f(a,b,c) = bc(b2 - c2) and, for variable x : y : z, the cubics u, v, w are given by u(x,y,z) = x(y2 + z2), v = u(y,z,x), w = u(z,x,y).
Parry proved that X(368) lies on the anticomplement of the Kiepert hyperbola, this anticomplement being given by the trilinear equation a2(b2 - c2)(x2) + b2(c2 - a2)(y2) + c2(a2 - b2)(z2) = 0.
X(369) = TRISECTED PERIMETER POINT
Trilinears x : y : z (see below)Barycentrics ax : by : cz
There exist points A', B', C' on segments BC, CA, AB, respectively, such that A'C + CB' = B'A + AC' = C'B + BA' and the lines AA', BB', CC' concur in X(369). Yff found trilinears for X(369) in terms of the unique real root, r, of the cubic polynomial
2t3 - 3(a + b + c)t2 + (a2 + b2 + c2 + 8bc + 8ca + 8ab)t - (cb2 + ac2 + ba2 + 5bc2 + 5ca2 + 5ab2 + 9abc),
as follows: x = bc(r - c + a)(r - a + b). Although x(a,c,b) ≠ x(a,b,c), Yff states that a symmetric but more elaborate form for x can be obtained.
X(370) = EQUILATERAL CEVIAN TRIANGLE POINT
Trilinears (reasonable trilinears are sought)Barycentrics (reasonable barycentrics are sought)
A point P is an equilateral cevian triangle point if the cevian triangle of P is equilateral. Jiang Huanxin introduced this notion in 1997.
Jiang Huanxin and David Goering, Problem 10358* and Solution, "Equilateral cevian triangles," American Mathematical Monthly 104 (1997) 567-570 [proposed 1994].
X(371) = KENMOTU POINT (CONGRUENT SQUARES POINT)
Trilinears cos(A - π/4) : cos(B - π/4) : cos(C - π/4)= cos A + sin A : cos B + sin B : cos C + sin C
Barycentrics sin A cos(A - π/4) : sin B cos(B - π/4) : sin C cos(C - π/4)
There exist three congruent squares U, V, W positioned in ABC as follows: U has opposing vertices on segments AB and AC; V has opposing vertices on segments BC and BA; W has opposing vertices on segments CA and CB, and there is a single point common to U, V, W. The common point, X(371), may have first been published in Kenmotu's Collection of Sangaku Problems in 1840, indicating that its first appearance may have been anonymously inscribed on a wooden board hung up in a Japanese shrine or temple. (The Kenmotu configuration uses only half-squares; i.e., isosceles right triangles). Trilinears were found by John Rigby.
The edgelength of the three squares is 21/2abc/(a2 + b2 + c2 + 4σ), where σ = area(ABC). (Edward Brisse, 2/12/00)
Hidetoshi Fukagawa, Traditional Japanese Mathematics Problems of the 18th and 19th Centuries, forthcoming.
Floor van Lamoen, Vierkanten in een driehoik: 3. Congruente vierkanten
Tony Rothman, with the cooperation of Hidetoshi Fukagawa, Japanese Temple Geometry (feature article in Scientific American
X(371) lies on these lines:
2,486 3,6 4,485 25,493 140,615 193,488 315,491 492,641 601,606 602,605
X(371) = reflection of X(372) about X(32)
X(371) = isogonal conjugate of X(485)
X(371) = inverse of X(372) in the Brocard circle
X(371) = complement of X(637)
X(371) = anticomplement of X(639)
X(371) = X(4)-Ceva conjugate of X(372)
X(372) = HARMONIC CONJUGATE OF X(371) WRT X(3) AND X(6)
Trilinears cos(A + π/4) : cos(B + π/4) : cos(C + π/4)= cos A - sin A : cos B - sin B : cos C - sin C
Barycentrics sin A cos(A + π/4) : sin B cos(B + π/4) : sin C cos(C + π/4)
For details and references, see X(371).
X(372) lies on these lines:
2,485 3,6 4,486 25,494 193,487 315,492 601,605 602,606
X(372) = reflection of X(371) about X(32)
X(372) = isogonal conjugate of X(486)
X(372) = inverse of X(371) in the Brocard circle
X(372) = complement of X(638)
X(372) = anticomplement of X(640)
X(372) = X(4)-Ceva conjugate of X(371)
X(373) = CENTROID OF THE PEDAL TRIANGLE OF THE CENTROID
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2bc + ac cos C + ab cos B= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a(b4 + c4 - a2b2 - a2c2 - 6b2c2)
Barycentrics h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = 2abc + ca2cos C + ba2cos B
X(373) lies on these lines: 2,51 5,113 110,575 181,748 216,852 354,375
Let X = X(373) and let V be the vector-sum XA + XB + XC; then V = X(2)X(51)
X(374) = CENTROID OF THE PEDAL TRIANGLE OF X(9)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2b + 2c - 3a + (c + a)cos C + (b + a)cos BBarycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(374) lies on these lines: 6,354 9,517 44,65 51,210
X(375) = CENTROID OF THE PEDAL TRIANGLE OF X(10)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2bc(b + c) + ca(c + a)cos C + ab(a + b)cos BBarycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(375) lies on these lines: 44,181 51,210 354,373
X(375) = midpoint between X(51) and X(210)
X(376) = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2)
Trilinears 5 cos A - cos(B - C) : 5 cos B - cos(C - A) : 5 cos C - cos(A - B)= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)(5 sin 2A - sin 2B - sin 2C)
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 5 sin 2A - sin 2B - sin 2C
X(376) is the reflection of X(2) about X(3).
X(376) lies on these lines:
1,553 2,3 35,388 36,497 40,519 55,1056 56,1058 69,74 98,543 103,544 104,528 110,541 112,577 165,515 316,1007 390,999 476,841 477,691 487,490 488,489 516,551
X(376) = midpoint between X(2) and X(20)
X(376) = reflection of X(I) about X(J) for these (I,J): (2,3), (4,2)
X(376) = anticomplement of X(381)
Let X = X(376) and let V be the vector-sum XA + XB + XC; then V = X(382)X(3) = X(4)X(20).
X(377)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b4 + c4 - a4 - 2b2c2 - 2abc(a + b + c))Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b4 + c4 - a4 - 2b2c2 - 2abc(a + b + c)
X(377) lies on these lines:
1,224 2,3 7,8 10,46 78,226 81,387 142,950 145,1056 149,1058 225,1038 274,315 908,936 1060,1068
X(377) = anticomplement of X(405)
X(378) = HARMONIC CONJUGATE OF X(24) WRT X(3) AND X(4)
Trilinears sec A + 2 cos A : sec B + 2 cos B : sec C + 2 cos CBarycentrics tan A + sin 2A : tan B + sin 2B : tan C + sin 2C
X(378) lies on these lines:
1,1063 2,3 6,74 33,36 34,35 54,64 99,264 185,578 232,574 477,935 847,1105
X(378) = reflection of X(I) about X(J) for these (I,J): (4,427), (22,3)
X(378) = inverse of X(403) in the orthocentroidal circle
X(379)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a5 + (b + c)(bca2 - (bc + ca + ab)(b - c)2)]Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a5 + (b + c)(bca2 - (bc + ca + ab)(b - c)2
X(379) lies on these lines: 2,3 6,7 41,226 63,169 264,823
X(379) = inverse of X(857) in the orthocentroidal circle
X(380)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)[3a3 + (b + c)(3a2 + (b - c)2 + a(b + c))]Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(380) lies on these lines: 1,19 6,40 9,55 165,579 223,608 281,950 282,1036
X(381) = MIDPOINT OF X(2) BETWEEN X(4)
Trilinears 2 cos(B - C) - cos A : 2 cos(C - A) - cos B : 2 cos(A - B) - cos C= cos A + 4 cos B cos C : cos B + 4 cos C cos A : cos C + 4 cos A cos B
Barycentrics a(cos A + 4 cos B cos C) : b(cos B + 4 cos C cos A) : c(cos C + 4 cos A cos B)
X(381) lies on these lines:
2,3 6,13 11,999 49,578 51,568 54,156 98,598 114,543 118,544 119,528 125,541 127,133 155,195 183,316 184,567 210,517 262,671 264,339 298,622 299,621 302,616 303,617 355,519 388,496 495,497 511,599 515,551
X(381) = midpoint between X(2) and X(4)
X(381) = reflection of X(I) about X(J) for these (I,J): (2,5), (3,2)
X(381) = complement of X(376)
X(381) = anticomplement of X(549)
Let X = X(381) and let V be the vector-sum XA + XB + XC; then V = X(20)X(3) = X(3)X(4) = X(185)X(52) = X(399)X(146) = X(74)X(265) = X(40)X(355) = X(376,381) = X(4,382).
X(382) = REFLECTION OF CIRCUMCENTER ABOUT ORTHOCENTER
Trilinears cos A - 4 cos B cos C : cos B - 4 cos C cos A : cos C - 4 cos A cos BBarycentrics a(cos A - 4 cos B cos C) : b(cos B - 4 cos C cos A) : c(cos C - 4 cos A cos B)
X(382) lies on these lines: 2,3 64,265 155,399 185,568 195,1078 355,516 952,962
X(382) = reflection of X(I) about X(J) for these (I,J): (3,4), (20,5)
X(382) = inverse of X(546) in the orthocentroidal circle
X(382) = anticomplement of X(550)
X(383)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = csc(B - C) [sin 2B cos(C - ω) sin(C + π/3) - sin 2C cos(B - ω) sin(B + π/3)]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(383) lies on these lines: 2,3 13,262 14,98 183,621 299,511 325,622
X(383) = inverse of X(1080) in the orthocentroidal circle
X(384)
Trilinears bc(a4 + b2c2) : ca(b4 + c2a2) : ab(c4 + a2b2)Barycentrics a4 + b2c2 : b4 + c2a2 : c4 + a2b2
A center on the Euler line; contributed by John Conway, email, 1998.
X(384) lies on these lines:
1,335 2,3 6,194 32,76 39,83 141,1031 172,350 185,287 316,626 694,695
X(384) = isogonal conjugate of X(695)
X(384) = eigencenter of anticevian triangle of X(385)
X(385) = HARMONIC CONJUGATE OF X(384) WRT X(32) AND X(76)
Trilinears bc(a4 - b2c2) : ca(b4 - c2a2) : ab(c4 - a2b2)Barycentrics a4 - b2c2 : b4 - c2a2 : c4 - a2b2
Contributed by John Conway, 1998.
X(385) lies on these lines:
1,257 2,6 3,194 23,523 30,148 32,76 55,192 56,330 98,511 99,187 111,892 115,316 171,894 232,648 248,290 251,308 262,576
X(385) = reflection of X(I) about X(J) for these (I,J): (99,187), (298,395), (299,396), (316,115), (325,230)
X(385) = isogonal conjugate of X(694)
X(385) = anticomplement of X(325)
X(385) = X(I)-Ceva conjugate of X(J) for these (I,J): (98,2), (511,401)
X(385) = crosspoint of X(290) and X(308)
X(385) = X(I)-Hirst inverse of X(J) for these (I,J): (2,6), (3,194)
X(386)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 + bc + ca + ab)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 + c2 + bc + ca + ab)
X(386) lies on these lines:
1,2 3,6 31,35 40,1064 55,595 56,181 57,73 65,994 81,404 474,940 758,986 872,984
X(386) = inverse of X(58) in the Brocard circle
X(387)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[-a4 + 2a2(a + b + c)2 + (b2 - c2)2]Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = -a4 + 2a2(a + b + c)2 + (b2 - c2)2
X(387) lies on these lines:
1,2 4,6 20,58 40,579 65,278 81,377 390,595 443,940
X(388)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a2 + (b + c)2]/(b + c -a)= 1 + cos B cos C : 1 + cos C cos A : 1 + cos A cos B
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2 + (b + c)2]/(b + c - a)
X(388) lies on these lines:
1,4 2,12 3,495 5,999 7,8 10,57 11,153 20,55 29,1037 35,376 36,498 79,1000 108,406 171,603 201,984 329,960 354,938 355,942 381,496 442,956 452,1001 612,1038 750,1106 1059,1067
X(388) = isogonal conjugate of X(1036)
X(388) = anticomplement of X(958)
X(389) = CENTER OF THE TAYLOR CIRCLE
Trilinears cos A - cos 2A cos(B - C) : cos B - cos 2B cos(C - A) : cos C - cos 2C cos(A - B)Barycentrics a[cos A - cos 2A cos(B - C)] : b[cos B - cos 2B cos(C - A)] : c[cos C - cos 2C cos(A - B)]
If ABC is acute then X(389) is the Spieker center of the orthic triangle. Peter Yff reports (Sept. 19, 2001) that since X(389) is on the Brocard axis, there must exist T for which X(389) is sin(A+T) : sin(B+T) : sin(C+T), and that tan(T) = - cot A cot B cot C.
X(389) lies on these lines:
3,6 4,51 24,184 30,143 54,186 115,129 217,232 517,950
X(389) = midpoint between X(I) and X(J) for these (I,J): (3,52), (4,185)
X(389) = inverse of X(578) in the Brocard circle
X(389) = crosspoint of X(4) and X(54)
X(390) REFLECTION OF GERGONNE POINT ABOUT INCENTER
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)[3a2 + (b - c)2]Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c - a)[3a2 + (b - c)2]
X(390) = lies on these lines:
1,7 2,11 3,1058 4,495 8,9 30,1056 40,938 144,145 376,999 387,595 496,631 944,971 952,1000
X(390) = midpoint between X(144) and X(145)
X(390) = reflection of X(I) about X(J) for these (I,J): (7,1), (8,9)
X(391)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(3a + b + c)(b + c - a)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (3a + b + c)(b + c - a)
X(391) lies on these lines:
2,6 8,9 20,573 37,145 75,144 319,344
X(392)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b2 + c2 - a2) + 4abcBarycentrics af(a,b,c): bf(b,c,a): cf(c,a,b)
X(392) lies on these lines:
1,6 2,517 8,1000 10,11 21,104 40,474 55,997 63,999 78,1057 210,519 329,1056 354,551 442,946 443,962 452,944 495,908
Let X = X(392) and let V be the vector-sum XA + XB + XC; then V = X(65)X(1) = X(8)X(72).
X(393)
Trilinears bc tan2A : bc tan2B : bc tan2CBarycentrics tan2A : tan2B : tan2C
X(393) lies on these lines:
1,836 2,216 4,6 19,208 20,577 24,254 25,1033 27,967 33,42 37,158 69,297 107,111 193,317 230,459 278,1108 342,948 394,837 800,1093
X(393) = cevapoint of X(4) and X(459)
X(393) = X(25)-cross conjugate of X(4)
X(394)
Trilinears cos A cot A : cos B cot B : cos C cot CBarycentrics cos2A : cos2B : cos2C
X(394) lies on these lines: 2,6 3,49 20,1032 22,110 25,511 63,77 72,1060 76,275 78,271 287,305 297,315 329,651 393,837 399,541 470,633 471,634 472,622 473,621 611,612 613,614 1062,1069
X(394) = X(69)-Ceva conjugate of X(3)
X(394) = crosspoint of X(493) and X(494)
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(394); then W = X(25)X(394).
X(395) = MIDPOINT OF X(14) AND X(16)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(B - C) + 2 cos(A + π/3)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,B,A)
X(395) lies on these lines:
2,6 3,398 5,13 14,16 15,549 39,618 53,472 61,140 115,530 187,531 202,495 216,465 466,577 532,624 533,619
X(395) = reflection of X(396) about X(230)
X(395) = midpoint between X(I) and X(J) for these (I,J): (14,16), (298,385)
X(395) = complement of X(299)
X(395) = crosspoint of X(2) and X(14)
X(396) = MIDPOINT OF X(13) AND X(15)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(B - C) + 2 cos(A - π/3)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,B,A)
X(396) lies on these lines:
2,6 3,397 5,14 13,15 16,549 39,619 53,473 62,140 115,531 187,530 203,495 216,466 465,577 532,618 533,623
X(396) = midpoint between X(I) and X(J) for these (I,J): (13,15), (299,385)
X(396) = reflection of X(395) about X(230)
X(396) = anticomplement of X(298)
X(396) = crosspoint of X(2) and X(13)
X(397) CROSSPOINT OF ORTHOCENTER AND 1st NAPOLEON POINT
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(B - C) - 2 cos(A + π/3)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,B,A)
X(397) lies on these lines: 3,396 4,6 5,13 14,546 15,550 16,17 30,61 51,462 141,634 184,463 202,496 524,633 532,635
X(397) = crosspoint of X(4) and X(17)
X(398) CROSSPOINT OF ORTHOCENTER AND 2nd NAPOLEON POINT
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(B - C) - 2 cos(A - π/3)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,B,A)
X(398) lies on these lines:
3,395 4,6 5,14 13,546 15,18 16,550 30,62 51,463 141,633 184,462 203,496 524,634 533,636
X(398) = crosspoint of X(4) and X(18)
X(399) = PARRY REFLECTION POINT
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = 5 cos A - 4 cos B cos C - 8 sin B sin C cos2A
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,B,A)
Let L, M, N be lines through A, B, C, respectively, parallel to the Euler line. Let L' be the reflection of L about sideline BC, let M' be the reflection of M about sideline CA, and let N' be the reflection of N about sideline AB. The lines L', M', N' concur in X(399).
Cyril Parry, Problem 10637, American Mathematical Monthly 105 (1998) 68.
X(399) lies on these lines:
3,74 4,195 6,13 30,146 155,382 394,541
X(399) = reflection of X(I) about X(J) for these (I,J): (3,110), (265,113)
X(399) = X(I)-Ceva conjugate of X(J) for these (I,J): (30,3), (323,6)
Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(399); then W = X(74)X(399).
X(400) = YFF-MALFATTI POINT
Trilinears csc4(A/4) : csc4(B/4) : csc4(C/4)Barycentrics sin A csc4(A/4) : sin B csc4(B/4) : sin C csc4(C/4)
In 1997, Yff considered the configuration for the 1st Ajima-Malfatti point, X(179). He proved that the same tangencies are possible in another way if the circles are not required to lie inside ABC. With tangency points labeled as before, the lines AA', BB', CC' concur in X(400).