PART 3
X(401) = BAILEY POINT
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = [sin 2B sin 2C - sin2(2A)](csc A)
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin 2B sin 2C - sin2(2A)
X(401) lies on these lines:
2,3 50,338 97,276 248,290 264,577 287,511 323,525
X(401) = anticomplement of X(297)
X(401) = X(I)-Ceva conjugate of X(J) for these (I,J): (287,2), (511,385)
X(401) = crosspoint of X(276) and X(290)
X(401) = X(2)-Hirst inverse of X(3)
X(402) = GOSSARD PERSPECTOR
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = p(a,b,c)y(a,b,c)/a, polynomials p and y as given below
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = p(a,b,c)y(a,b,c), polynomials p and y as given below
In A History of Mathematics, Florian Cajori wrote, "H. C. Gossard of the University of Oklahoma showed in 1916 that the three Euler lines of the triangles formed by the Euler line and the sides, taken by twos, of a given triangle, form a triangle . . . perspective with the given triangle and having the same Euler line." Let ABC be the given triangle and A'B'C' the Gossard triangle - that is, the triangle perspective with the given triangle and having the same Euler line. The lines AA', BB', CC' concur in X(402), named the Gosssard perspector by John Conway (1998). Barycentrics for X(402) were received from Paul Yiu (2/20/99); the polynomials p and y referred to above are given as follows:
p(a,b,c) = 2a4 - a2b2 - a2c2 - (b2 - c2)2
y(a,b,c) = a8 - a6(b2 + c2) + a4(2b2 - c2)(2c2 - b2) + [(b2 - c2)2][3a2(b2 + c2) - b4 - c4 - 3b2c2]
X(402) lies on this line: 2,3
X(403) = X(36) OF THE ORTHIC TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)(1 + cos 2B + cos 2C)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (tan A)(1 + cos 2B + cos 2C)
X(403) lies on these lines: 2,3 112,230 115,232 847,1093
X(403) = midpoint between X(4) and X(186)
X(403) = reflection of X(186) about X(468)
X(403) = inverse of X(24) in the circumcircle
X(403) = inverse of X(4) in the nine-point circle
X(403) = inverse of X(378) in the orthocentroidal circle
X(403) = X(113)-cross conjugate of X(4)
X(404) = HARMONIC CONJUGATE OF X(21) WRT X(2) AND X(3)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a + b + c) - a(b2 + c2 - a2)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = abc(a + b + c) - a2(b2 + c2 - a2)
X(404) lies on these lines:
1,88 2,3 8,56 10,36 31,978 46,997 57,78 60,662 63,936 69,1014 81,386 104,355 108,318 145,999 149,496 603,651 612,988 976,982
X(405)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a + b + c) + abc cos ABarycentrics b + c + (1 + a)cos A : c + a + (1 + b)cos B : a + b + (1 + c)cos C
X(405) lies on these lines: 1,6 2,3 8,943 10,55 56,226 58,940 63,942 284,965 329,999 756,976 846,986
X(405) = inverse of X(442) in the orthocentroidal circle
X(405) = complement of X(377)
X(406)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a + b + c) + abc sec ABarycentrics b + c + a(1 + sec A) : c + a + b(1 + sec B) : a + b + c(1 + sec C)
X(406) lies on these lines:
2,3 8,1061 10,33 37,158 92,1068 108,388 208,226 261,317
X(406) inverse of X(475) in the orthocentroidal circle
X(407)
Trilinears (v + w)sec A : (w + u)sec B : (u + v)sec C, whereu : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)
Barycentrics (v + w)tan A : (w + u)tan B : (u + v)tan C
X(407) lies on these lines: 2,3 12,228 65,225 117,136
X(407) = crosspoint of X(4) and X(225)
X(408)
Trilinears (v + w)cos A : (w + u)cos B : (u + v)cos C, whereu : v : w = X(29); e.g., u = u(A,B,C) = (sec A)/(cos B + cos C)
Barycentrics (v + w)sin 2A : (w + u)sin 2B : (u + v)sin 2C
X(408) lies on these lines: 2,3 73,228
X(409)
Trilinears u2 + vw : v2 + wu : w2 + uv, whereu : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)
Barycentrics a(u2 + vw) : b(v2 + wu) : c(w2 + uv)
X(409) lies on these lines: 2,3 65,1098
X(410)
Trilinears u2 + vw : v2 + wu : w2 + uv, whereu : v : w = X(29); e.g., u = u(A,B,C) = (sec A)/(cos B + cos C)
Barycentrics a(u2 + vw) : b(v2 + wu) : c(w2 + uv)
X(410) lies on this line: 2,3
X(411)
Trilinears (1/v)2 + (1/w)2 - (1/u)2 : (1/w)2 + (1/u)2 - (1/v)2 : (1/u)2 + (1/v)2 - (1/w)2, whereu : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)
Barycentrics a[(1/v)2 + (1/w)2 - (1/u)2] : b[(1/w)2 + (1/u)2 - (1/v)2] : c[(1/u)2 + (1/v)2 - (1/w)2]
X(411) lies on these lines: 2,3 35,516 40,78 55,962 81,581 165,936 185,970 243,821 255,651
X(412)
Trilinears (1/v)2 + (1/w)2 - (1/u)2 : (1/w)2 + (1/u)2 - (1/v)2 : (1/u)2 + (1/v)2 - (1/w)2, whereu : v : w = X(29); e.g., u = u(A,B,C) = (sec A)/(cos B + cos C)
Barycentrics a[(1/v)2 + (1/w)2 - (1/u)2] : b[(1/w)2 + (1/u)2 - (1/v)2] : c[(1/u)2 + (1/v)2 - (1/w)2]
X(412) lies on these lines: 2,3 40,92 46,158 63,318 65,243 162,580 225,775 278,962
X(413)
Trilinears u3(v2 + w2 - vw) : v3(w2 + u2 - wu) : w3(u2 + v2 - uv), whereu : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)
Barycentrics au3(v2 + w2 - vw) : bv3(w2 + u2 - wu) : cw3(u2 + v2 - uv)
X(413) lies on this line: 2,3
X(414)
Trilinears u3(v2 + w2 - vw) : v3(w2 + u2 - wu) : w3(u2 + v2 - uv), whereu : v : w = X(29); e.g., u = u(A,B,C) = (sec A)1/(cos B + cos C)
Barycentrics au3(v2 + w2 - vw) : bv3(w2 + u2 - wu) : cw3(u2 + v2 - uv)
X(414) lies on this line: 2,3
X(415)
Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, whereu : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)
Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
X(415) lies on these lines: 2,3 162,238
X(415) = X(4)-Hirst inverse of X(29)
X(416)
Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, whereu : v : w = X(29); e.g., u = u(A,B,C) = (sec A)/(cos B + cos C)
Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
X(416) lies on this line: 2,3
X(416) = X(3)-Hirst inverse of X(21)
X(417)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(sec2B + sec2C)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin 2A)(sec2B + sec2C)
X(417) lies on this line: 2,3
X(417) = X(3)-Ceva conjugate of X(185)
X(418)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(csc 2B + csc 2C)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin 2A)(csc 2B + csc 2C)
X(418) lies on these lines: 2,3 51,216 97,110 154,160 157,161 184,577
X(418) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,216), (216,217)
X(419)
Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, whereu : v : w = X(31); e.g., u = u(A,B,C) = a2
Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
X(419) lies on these lines: 2,3 238,242
X(419) = X(4)-Hirst inverse of X(25)
X(420)
Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, whereu : v : w = X(38); e.g., u = u(a,b,c) = b2 + c2
Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
X(420) lies on this line: 2,3
X(420) = X(4)-Hirst inverse of X(427)
X(421)
Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, whereu : v : w = X(47); e.g., u = u(A,B,C) = cos 2A
Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
X(421) lies on this line: 2,3
X(421) = X(4)-Hirst inverse of X(24)
X(422)
Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, whereu : v : w = X(58); e.g., u = u(a,b,c) = a/(b + c)
Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
X(422) lies on these lines: 2,3 162,242
X(422) = X(4)-Hirst inverse of X(28)
X(423)
Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, whereu : v : w = X(81); e.g., u = u(a,b,c) = 1/(b + c)
Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
X(423) lies on this line: 2,3
X(423) = X(4)-Hirst inverse of X(27)
X(424)
Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, whereu : v : w = X(191); e.g., u = u(a,b,c) = (b + c - a)(bc + ca + ab) + b3 + c3 - a3
Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
X(424) lies on this line: 2,3
X(424) = X(4)-Hirst inverse of X(451)
X(425)
Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, whereu : v : w = X(283); e.g., u = u(A,B,C) = (cos A)/(cos B + cos C)
Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
X(425) lies on this line: 2,3 243,662
X(425) = X(4)-Hirst inverse of X(21)
X(426)
Trilinears (v2 + w2)cos A : (w2 + u2)cos B : (u2 + v2)cos C, whereu : v : w = X(19); e.g., u = u(A,B,C) = tan A
Barycentrics (v2 + w2)sin 2A : (w2 + u2)sin 2B : (u2 + v2)sin 2C
X(426) lies on this line: 2,3
X(427)
Trilinears (v + w)sec A : (w + u)sec B : (u + v)sec C, whereu : v : w = X(31); e.g., u = u(a,b,c) = a2
Barycentrics (v + w)tan A : (w + u)tan B : (u + v)tan C
X(427) lies on these lines:
2,3 6,66 11,33 12,34 51,125 53,232 98,275 112,251 114,136 183,317 230,571 264,305 343,511
X(427) = midpoint between X(4) and X(378)
X(427) = inverse of X(468) in the nine-point circle
X(427) = inverse of X(25) in the orthocentroidal circle
X(427) = complement of X(22)
X(427) = X(112)-Ceva conjugate of X(523)
X(427) = X(39)-cross conjugate of X(141)
X(427) = crosspoint of X(4) and X(264)
X(427) = X(4)-Hirst inverse of X(429)
X(428)
Trilinears (v + w)sec A : (w + u)sec B : (u + v)sec C, whereu : v : w = X(38); e.g., u = u(a,b,c) = b2 + c2
Barycentrics (v + w)tan A : (w + u)tan B : (u + v)tan C
X(428) lies on these lines: 2,3 132,137
X(429)
Trilinears (v + w)sec A : (w + u)sec B : (u + v)sec C, whereu : v : w = X(58); e.g., u = u(a,b,c) = a/(b + c)
Barycentrics (v + w)tan A : (w + u)tan B : (u + v)tan C
X(429) lies on these lines: 2,3 11,1104 12,37 108,961 119,136 495,1068
X(429) = X(108)-Ceva conjugate of X(523)
X(430)
Trilinears (v + w)sec A : (w + u)sec B : (u + v)sec C, whereu : v : w = X(81); e.g., u = u(a,b,c) = 1/(b + c)
Barycentrics (v + w)tan A : (w + u)tan B : (u + v)tan C
X(430) lies on these lines: 2,3 118,136 210,594
X(431)
Trilinears (v + w)sec A : (w + u)sec B : (u + v)sec C, whereu : v : w = X(283); e.g., u = u(A,B,C) = (cos A)/(cos B + cos C)
Barycentrics (v + w)tan A : (w + u)tan B : (u + v)tan C
X(431) lies on these lines: 2,3 119,135
X(432)
Trilinears (v2 + w2)sec A : (w2 + u2)sec B : (u2 + v2)sec C, whereu : v : w = X(155); e.g., u = u(A,B,C) = (cos A)(cos2B + cos2C - cos2A)
Barycentrics (v2 + w2)tan A : (w2 + u2)tan B : (u2 + v2)tan C
X(432) lies on this line: 2,3
X(433)
Trilinears (v2 + w2)sec A : (w2 + u2)sec B : (u2 + v2)sec C, where u : v : w = X(159)Barycentrics (v2 + w2)tan A : (w2 + u2)tan B : (u2 + v2)tan C
X(433) lies on this line: 2,3
X(434)
Trilinears (v2 + w2)sec A : (w2 + u2)sec B : (u2 + v2)sec C, where u : v : w = X(195)Barycentrics (v2 + w2)tan A : (w2 + u2)tan B : (u2 + v2)tan C
X(434) lies on this line: 2,3
X(435)
Trilinears (v2 + w2)sec A : (w2 + u2)sec B : (u2 + v2)sec C, where u : v : w = X(399)Barycentrics (v2 + w2)tan A : (w2 + u2)tan B : (u2 + v2)tan C
X(435) lies on this line: 2,3
X(436)
Trilinears (u2 + vw)sec A : (v2 + wu)sec B : (w2 + uv)sec C, whereu : v : w = X(48); e.g., u(A,B,C) = sin 2A
Barycentrics (u2 + vw)tan A : (v2 + wu)tan B : (w2 + uv)tan C
X(436) lies on these lines: 2,3 51,107 110,324 578,1093
X(437)
Trilinears (u2 + vw)sec A : (v2 + wu)sec B : (w2 + uv)sec C, where u : v : w = X(214)Barycentrics (u2 + vw)tan A : (v2 + wu)tan B : (w2 + uv)tan C
X(437) lies on this line: 2,3
X(438)
Trilinears (u2 + vw)csc A : (v2 + wu)csc B : (w2 + uv)csc C, whereu : v : w = X(204); e.g., u(A,B,C) = (tan A)(tan B + tan C - tan A)
Barycentrics u2 + vw : v2 + wu : w2 + uv
X(438) lies on this line: 2,3
X(439)
Trilinears au2 : bv2 : cw2, whereu : v : w = X(193); e.g., u(A,B,C) = (csc A)(cot B + cot C - cot A)
Barycentrics (au)2 : (bv)2 : (cw)2
X(439) lies on this line: 2,3
X(439) = X(459)-Ceva conjugate of X(193)
X(440)
Trilinears bc(v + w) : ca(w + u) : ab(u + v), whereu : v : w = X(28); e.g., u(a,b,c) = (tan A)/(b + c)
Barycentrics v + w : w + u : u + v
X(440) lies on these lines: 2,3 37,226 72,306 118,122 950,1104
X(440) = complement of X(27)
X(440) = X(190)-Ceva conjugate of X(525)
X(440) = crosspoint of X(2) and X(306)
X(441)
Trilinears bc(v + w) : ca(w + u) : ab(u + v), whereu : v : w = X(240); e.g., u(A,B,C) = sec A cos(A + ω)
Barycentrics v + w : w + u : u + v
X(441) lies on these lines: 2,3 141,577 525,647
X(441) = complement of X(297)
X(442) COMPLEMENT OF SCHIFFLER POINT
Trilinears bc(v + w) : ca(w + u) : ab(u + v), whereu : v : w = X(284); e.g., u(A,B,C) = (sin A)/(cos B + cos C)
Barycentrics v + w : w + u : u + v
X(442) lies on these lines: 2,3 8,495 9,46 10,12 11,214 100,943 115,120 119,125 274,325 388,956 392,946
X(442) = midpoint between X(79) and X(191)
X(442) = inverse of X(405) in the orthocentroidal circle
X(442) = complement of X(21)
X(442) = X(100)-Ceva conjugate of X(523)
X(442) = crosspoint of X(264) and X(321)
X(443)
Trilinears bc(v + w) : ca(w + u) : ab(u + v), whereu : v : w = X(380)
Barycentrics v + w : w + u : u + v
X(443) lies on these lines: 1,142 2,3 7,72 8,942 10,57 69,274 226,936 278,1038 387,940 392,962 579,966
X(443) = complement of X(452)
X(444)
Trilinears (v + w)tan A : (w + u)tan B : (u + v)tan C, whereu : v : w = X(256); e.g., u(a,b,c) = 1/(a2 + bc)
Barycentrics (v + w)(sin A tan A) : (w + u)(sin B tan B) : (u + v)(sin C tan C)
X(444) lies on these lines: 2,3 19,232
X(445)
Trilinears (v + w)csc 2A : (w + u)csc 2B : (u + v)csc 2C, whereu : v : w = X(79); e.g., u(a,b,c) = 1/(1 + 2 cos A)
Barycentrics (v + w)sec A : (w + u)sec B : (u + v)sec C
X(445) lies on this line: 2,3
X(446)
Trilinears u(v2 + w2) : v(w2 + u2) : w(u2 + v2), whereu : v : w = X(98); e.g., u(A,B,C) = sec(A + ω)
Barycentrics au(v2 + w2) : bv(w2 + u2) : cw(u2 + v2)
X(446) lies on this line: 2,3
X(446) = crosspoint of X(98) and X(511)
X(447)
Trilinears bc(u2 - vw) : ca(v2 - wu) : ab(w2 - uv), whereu : v : w = X(28); e.g., u(a,b,c) = (tan A)/(b + c)
Barycentrics u2 - vw : v2 - wu : w2 - uv
X(447) lies on this line: 2,3 340,540 350,811 519,648
X(447) = X(2)-Hirst inverse of X(27)
X(448)
Trilinears bc(u2 - vw) : ca(v2 - wu) : ab(w2 - uv), whereu : v : w = X(284); e.g., u(A,B,C) = (sin A)/(cos B + cos C)
Barycentrics u2 - vw : v2 - wu : w2 - uv
X(448) lies on this line: 2,3
X(448) = X(2)-Hirst inverse of X(21)
X(449)
Trilinears bc(u2 - vw) : ca(v2 - wu) : ab(w2 - uv), where u : v : w = X(380)Barycentrics u2 - vw : v2 - wu : w2 - uv
X(449) lies on this line: 2,3
X(449) = X(2)-Hirst inverse of X(452)
X(450) X(3)-HIRST INVERSE OF X(4)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)[cos4A - (cos B cos C)2]= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cos A)[sec4A - (sec B sec C)2]
Barycentrics h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = (tan A)(cos4A - (cos B cos C)2]
X(450) lies on these lines: 2,3 107,511 155,1075 1092,1093
X(450) = X(3)-Hirst inverse of X(4)
X(451)
Trilinears u sec A : v sec B : w sec C, where u : v : w = X(191)Barycentrics u tan A : v tan B : w tan C
X(451) lies on these lines: 2,3 12,108 281,1068
X(451) = X(4)-Hirst inverse of X(424)
X(452)
Trilinears u csc A : v csc B : w csc C, where u : v : w = X(380)Barycentrics u : v : w
X(452) lies on these lines: 1,329 2,3 8,9 34,347 63,938 72,145 388,1001 392,944 497,958 956,1058
X(452) = anticomplement of X(443)
X(452) = X(2)-Hirst inverse of X(449)
X(453)
Trilinears u2/(cos B + cos C) : v2/(cos C + cos A) : w2/(cos A + cos B), where u : v : w = X(46)Barycentrics (u2sin A)/(cos B + cos C) : (v2sin B)/(cos C + cos A) : (w2sin C)/(cos A + cos B)
X(453) lies on this line: 2,3
X(454)
Trilinears u2sec A : v2sec B : w2sec C, whereu : v : w = X(155); e.g., u(A,B,C) = (cos A)[cos2B + cos2C - cos2A]
Barycentrics u2tan A : v2tan B : w2tan C
X(454) lies on this line: 2,3
X(455)
Trilinears u2sec A : v2sec B : w2sec C, where u : v : w = X(159)Barycentrics u2tan A : v2tan B : w2tan C
X(455) lies on this line: 2,3
X(456)
Trilinears u2sec A : v2sec B : w2sec C, where u : v : w = X(195)Barycentrics u2tan A : v2tan B : w2tan C
X(456) lies on this line: 2,3
X(457)
Trilinears u2sec A : v2sec B : w2sec C, where u : v : w = X(399)Barycentrics u2tan A : v2tan B : w2tan C
X(457) lies on this line: 2,3
X(458)
Trilinears u csc 2A : v csc 2B : w csc 2C, whereu : v : w = X(182); e.g.; u(A,B,C) = cos(A - ω)
Barycentrics u sec A : v sec B : w sec C
X(458) lies on these lines: 2,3 6,264 76,275 141,317 239,318 273,894 315,343 340,599
X(458) = inverse of X(297) in the orthocentroidal circle
X(459)
Trilinears u tan A : v tan B : w tan C, whereu : v : w = X(193); e.g.; u(A,B,C) = (csc A)(cot B + cot C - cot A)
Barycentrics u sin A tan A : v sin B tan B : w sin C tan C
X(459) lies on these lines: 2,3 230,393 232,800 242,278 317,1007
X(459) = cevapoint of X(193) and X(439)
X(459) = X(393)-cross conjugate of X(4)
X(459) = X(4)-Hirst inverse of X(460)
X(460)
Trilinears u tan A : v tan B : w tan C, where u : v : w = X(230)Barycentrics u sin A tan A : v sin B tan B : w sin C tan C
X(460) lies on this line: 2,3
X(460) = X(4)-Hirst inverse of X(459)
X(461)
Trilinears u tan A : v tan B : w tan C, whereu : v : w = X(391); e.g., u(a,b,c) = bc(3a + b + c)(b + c - a)
Barycentrics u sin A tan A : v sin B tan B : w sin C tan C
X(461) lies on these lines: 2,3 33,200
X(462)
Trilinears u tan A : v tan B : w tan C, whereu : v : w = X(395); e.g., u(A,B,C) = cos(B - C) + 2 cos(A + π/3)
Barycentrics u sin A tan A : v sin B tan B : w sin C tan C
X(462) lies on these lines: 2,3 51,397 184,398
X(463)
Trilinears u tan A : v tan B : w tan C, whereu : v : w = X(396); e.g., u(A,B,C) = cos(B - C) + 2 cos(A - π/3)
Barycentrics u sin A tan A : v sin B tan B : w sin C tan C
X(463) lies on these lines: 2,3 51,398 184,397
X(464)
Trilinears u cot A : v cot B : w cot C, where u : v : w = X(387)Barycentrics u cos A : v cos B : w cos C
X(464) lies on these lines: 2,3 63,69
X(465)
Trilinears u cot A : v cot B : w cot C, whereu : v : w = X(397); e.g., u(A,B,C) = cos(B - C) - 2 cos(A + π/3)
Barycentrics u cos A : v cos B : w cos C
X(465) lies on these lines: 2,3 216,395 396,577
X(465) = complement of X(473)
X(466)
Trilinears u cot A : v cot B : w cot C, whereu : v : w = X(398); e.g., u(A,B,C) = cos(B - C) - 2 cos(A - π/3)
Barycentrics u cos A : v cos B : w cos C
X(466) lies on these lines: 2,3 216,396 395,577
X(446) = complement of X(472)
X(467)
Trilinears u csc 2A : v csc 2B : w csc 2C, whereu : v : w = X(52); e.g., u(A,B,C) = cos 2A cos(B - C)
Barycentrics u sec A : v sec B : w sec C
X(467) lies on these lines: 2,3 53,311
X(467) = X(317)-Ceva conjugate of X(52)
X(468) X(2)-LINE CONJUGATE OF X(3)
Trilinears u csc 2A : v csc 2B : w csc 2C, whereu : v : w = X(187); e.g., u(a,b,c) = a(2a2 - b2 - c2)
Barycentrics u sec A : v sec B : w sec C
X(468) lies on these lines: 2,3 98,685 107,842 111,935 230,231 250,325
X(468) = midpoint between X(186) and X(403)
X(468) = isogonal conjugate of X(895)
X(468) = inverse of X(25) in the circumcircle
X(468) = inverse of X(427) in the nine-point circle
X(468) = X(187)-cross conjugate of X(524)
X(468) = X(2)-line conjugate of X(3)
Let X = X(468) and let V be the vector-sum XA + XB + XC; then V = X(468)X(23).
X(469)
Trilinears u csc 2A : v csc 2B : w csc 2C, whereu : v : w = X(386); e.g., u(a,b,c) = a(b2 + c2 + bc + ca + ab)
Barycentrics u sec A : v sec B : w sec C
X(469) lies on these lines: 2,3 92,264 226,273
X(469) = inverse of X(27) in the orthocentroidal circle
X(470)
Trilinears sin(A + π/3) csc 2A : sin(B + π/3) csc 2B : sin(C + π/3) csc 2CBarycentrics sin(A + π/3) sec A : sin(B + π/3) sec B : sin(C + π/3) sec 2C
X(470) lies on these lines: 2,3 18,275 264,301 298,340 302,317 343,634 394,633
X(470) = inverse of X(471) in the orthocentroidal circle
X(470) = X(15)-cross conjugate of X(298)
X(470) = X(4)-Hirst inverse of X(471)
X(471)
Trilinears sin(A - π/3) csc 2A : sin(B - π/3) csc 2B : sin(C - π/3) csc 2CBarycentrics sin(A - π/3) sec A : sin(B - π/3) sec B : sin(C - π/3) sec 2C
X(471) lies on these lines: 2,3 17,275 264,300 299,340 303,317 343,633 394,634
X(471) = inverse of X(470) in the orthocentroidal circle
X(471) = X(16)-cross conjugate of X(299) for these (I,J): (71,10), (72,307), (440,2)
X(471) = X(4)-Hirst inverse of X(470)
X(472)
Trilinears cos(A + π/3) csc 2A : cos(B + π/3) csc 2B : cos(C + π/3) csc 2CBarycentrics cos(A + π/3) sec A : cos(B + π/3) sec B : cos(C + π/3) sec 2C
X(472) lies on these lines: 2,3 13,275 53,395 264,298 299,317 343,621 394,622
X(472) = inverse of X(473) in the orthocentroidal circle
X(472) = anticomplement of X(466)
X(472) = X(62)-cross conjugate of X(303)
X(473)
Trilinears cos(A - π/3) csc 2A : cos(B - π/3) csc 2B : cos(C - π/3) csc 2CBarycentrics cos(A - π/3) sec A : cos(B - π/3) sec B : cos(C - π/3) sec 2C
X(473) lies on these lines: 2,3 14,275 53,396 264,299 298,317 343,622 394,621
X(473) = inverse of X(472) in the orthocentroidal circle
X(473) = anticomplement of X(465)
X(473) = X(61)-cross conjugate of X(302)
X(474)
Trilinears cos A - (a + b + c)/a : cos B - (a + b + c)/b : cos C - (a + b + c)/cBarycentrics a cos A - (a + b + c) : b cos B - (a + b + c) : c cos C - (a + b + c)
X(474) lies on these lines: 2,3 8,999 10,56 35,1001 36,958 40,392 46,960 57,72 65,997 78,942 142,954 171,978 183,274 244,976 283,582 386,940 579,965 986,1054
X(475)
Trilinears sec A - (a + b + c)/a : sec B - (a + b + c)/b : sec C - (a + b + c)/cBarycentrics a sec A - (a + b + c) : b sec B - (a + b + c) : c sec C - (a + b + c)
X(475) lies on these lines: 2,3 8,1063 10,34 264,274 318,1068
X(475) = inverse of X(406) in the orthocentroidal circle
X(476) = TIXIER POINT
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[(1 + 2 cos 2A)(sin(B - C)]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
The reflection of X(110) about the Euler line; X(476) lies on the circumcircle. (Michel Tixier, 5/9/98). Also, X(476) is the center of the polar conic of X(30) with respect to the Neuberg cubic; this conic is a rectangular hyperbola passing through the incenter, the excenters, and X(30). (Peter Yff, 5/23/99)
X(476) lies on these lines: 2,842 3,477 23,94 30,74 99,850 110,523 111,230 250,933 376,841
X(476) = reflection of X(477) about X(3)
X(476) = isogonal conjugate of X(526)
X(476) = cevapoint of X(30) and X(523)
X(477) = TIXIER ANTIPODE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[4 cos A + cos(A - 2B) + cos(A - 2C) - 3 cos(B - C)]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
The reflection of X(476) about X(3), on the circumcircle. (Michel Tixier, 5/16/98)
X(477) lies on these lines: 3,476 30,110 50,112 74,523 107,186 376,691 378,935
X(477) = reflection of X(476) about X(3)
X(478) = CENTER OF YIU CONIC
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a4 - 2abc(b + c - a) - (b2 - c2)2]/(b + c - a)Barycentrics h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = af(a,b,c)
Center of the Yiu conic, which passes through the points outside the circumcircle at which the excircles of ABC are tangent to the sidelines of ABC. (PaulYiu, "The Clawson point and excircles," 1999)
X(478) lies on these lines: 6,19 9,1038 69,651 109,573 198,577 222,226
X(479)
Trilinears (tan A/2 sec A/2)2 : (tan B/2 sec B/2)2 : (tan C/2 sec C/2)2
Barycentrics tan3(A/2) : tan3(B/2) : tan3(C/2)
= 1/(b + c - a)3 : 1/(c + a - b)3 : 1/(a + b - c)3
Let A' be the point in which the incircle is tangent to a circle that passes through vertices B and C, and determine B' and C' cyclically. The lines AA', BB', CC' concur in X(479)
Clark Kimberling and Peter Yff, Problem 10678, American Mathematical Monthly 105 (1998) 666.
X(479) lies on these lines: 7,354 57,279 269,614
X(479) = isogonal conjugate of X(480)
X(479) = X(269)-cross conjugate of X(279)
X(480)
Trilinears (cot A/2 cos A/2)2 : (cot B/2 cos B/2)2 : (cot C/2 cos C/2)2Barycentrics (sin A)(cot A/2 cos A/2)2 : (sin B)(cot B/2 cos B/2)2 : (sin C)(cot C/2 cos C/2)2
The radical center of the three circles used to construct X(479). (Peter Yff, 5/6/98)
X(480) lies on these lines: 8,344 9,55 10,954 56,78 100,144
X(480) = isogonal conjugate of X(479)
X(480) = X(200)-Ceva conjugate of X(220)
X(481) = 1st EPPSTEIN POINT
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 - 2 sec A/2 cos B/2 cos C/2= 1 - 4(area)/[a(b + c - a)] : 1 - 4(area)/[b(c + a - b)] : 1 - 4(area)/[c(a + b - c) [E. Brisse, 3/20/01]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B).
Let S be the inner Soddy circle and Sa, Sb, Sc the Soddy circles tangent to S. Let Ia = S^Sa, Ea = Sb^Sc, and determine Ib, Ic, Eb, Ec cyclically. Then X(481) is the point of concurrence of lines Ia-to-Ea, Ib-to-Eb, Ic-to-Ec.
David Eppstein, "Tangent spheres and triangle centers," American Mathematical Monthly, 108 (2001) 63-66.
X(481) lies on these lines: 1,7 226,485
X(481) = X(79)-Ceva conjugate of X(482)
X(482) = 2nd EPPSTEIN POINT
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + 2 sec A/2 cos B/2 cos C/2= 1 + 4(area)/[a(b + c - a)] : 1 + 4(area)/[b(c + a - b)] : 1 + 4(area)/[c(a + b - c) [E. Brisse, 3/20/01]
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B).
Let S' be the outer Soddy circle and Sa, Sb, Sc the Soddy circles tangent to S. Let Ja = S'^Sa, Ea = Sb^Sc, and determine Jb, Jc, Eb, Ec cyclically. Then X(482) is the point of concurrence of lines Ja-to-Ea, Jb-to-Eb, Jc-to-Ec.
David Eppstein, "Tangent spheres and triangle centers," American Mathematical Monthly, 108 (2001) 63-66.
X(482) lies on these lines: 1,7 226,486
X(482) = X(79)-Ceva conjugate of X(481)
X(483) = RADICAL CENTER OF AJIMA-MALFATTI CIRCLES
Trilinears sec2A/4 : sec2B/4 : sec2C/4= 1/(1 + cos A/2) : 1/(1 + cos B/2) : 1/(1 + cos C/2)
Barycentrics sin A sec2A/4 : sin B sec2B/4 : sin C sec2C/4
The Ajima-Malfatti circles are described at X(179). (Peter Yff, 6/1/98)
X(483) lies on these lines: 8,178 173,180 174,175
X(484) = EVANS PERSPECTOR
Trilinears 1 + 2(cos A - cos B - cos C) : 1 + 2(cos B - cos C - cos A) : 1 + 2(cos C + cos A - cos B)Barycentrics a[1 + 2(cos A - cos B - cos C)] : b[1 + 2(cos B - cos C - cos A)] : c[1 + 2(cos C + cos A - cos B)]
X(484) is the perspector of the extriangle and the triangle A'B'C', where A' is the reflection of vertex A about sideline BC and B', C' are determined cyclically. (Lawrence Evans, 10/22/98)
X(484) lies on these lines: 1,3 10,191 12,79 30,80 63,535 100,758 499,962 759,901 1046,1048
X(484) = reflection of X(1) about X(36)
X(484) = inverse of X(35) in the circumcircle
X(484) = X(80)-Ceva conjugate of X(1)
Centers 485- 495,
371, and 372: Vierkanten in een driehoek - triangle centers associated with squares.
X(485) = VECTEN POINT
Trilinears sec(A - π/4) : sec(B - π/4) : sec(C - π/4)Barycentrics sin A sec(A - π/4) : sin B sec(B - π/4) : sin C sec(C - π/4)
X(485) is the perspector of triangles associated with squares that circumscribe ABC. For details, visit Floor van Lamoen's site, Vierkanten in een driehoek: 1. Omgeschreven vierkanten (van Lamoen, 4/26/98) and his article "Friendship Among Triangle Centers," Forum Geometricorum, 1 (2001) 1-6.
X(485) lies on these lines: 2,372 3,590 4,371 5,6 69,639 76,491 226,481 489,671
X(485) = isogonal conjugate of X(371)
X(485) = isotomic conjugate of X(492)
X(485) = complement of X(488)
X(485) = anticomplement of X(641)
X(485) = X(3)-cross conjugate of X(486)
X(486) = INNER VECTEN POINT
Trilinears sec(A + π/4) : sec(B + π/4) : sec(C + π/4)Barycentrics sin A sec(A + π/4) : sin B sec(B + π/4) : sin C sec(C + π/4)
X(486) is a perspector of triangles associated with squares that circumscribe ABC. For details and references, see X(485). (Floor van Lamoen, 4/26/98)
X(486) lies on these lines: 2,371 3,615 4,372 5,6 76,492 141,591 226,482 490,671
X(486) = isogonal conjugate of X(372)
X(486) = isotomic conjugate of X(491)
X(486) = complement of X(487)
X(485) = anticomplement of X(642)
X(486) = X(3)-cross conjugate of X(485)
X(487)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 - csc A sin B sin C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(487) is a perspector of triangles associated with squares that circumscribe ABC. (Floor van Lamoen, 4/29/98)
X(487) lies on these lines: 2,371 3,69 4,489 20,638 193,372 376,490 492,631
X(487) = anticomplement of X(486)
X(487) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,488), (489,20), (491,2)
X(488)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 + csc A sin B sin C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(488) is a perspector of triangles associated with squares that circumscribe ABC. For details, see Vierkanten in een driehoek: 2. Meer punten uit omgeschreven vierkanten (Floor van Lamoen, 4/29/98)
X(488) lies on these lines: 2,372 3,69 4,490 193,371 376,489 491,631
X(488) = anticomplement of X(485)
X(488) = X(I)-Ceva conjugate of X(J), for these (I,J): (4,487), (490,20), (492,2)
X(489)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 - csc A sin B sin C) - cos B cos CBarycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(489) is a perspector of triangles associated with squares that circumscribe ABC. For details and reference, see X(488). (Floor van Lamoen, 4/29/98)
X(489) lies on these lines: 3,492 4,487 20,64 30,638 176,664 376,488 485,671
X(489) = cevapoint of X(20) and X(487)
X(490)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 + csc A sin B sin C) - cos B cos CBarycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(490) is a perspector of triangles associated with squares that circumscribe ABC. For details and reference, see X(488). (Floor van Lamoen, 4/29/98)
X(490) lies on these lines: 3,491 4,488 20,64 30,637 175,664 376,487 486,671
X(490) = cevapoint of X(20) and X(488)
X(491)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 - csc A sin B sin C) + cos B cos CBarycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(491) is a pole associated with squares that circumscribe ABC. For details and reference, see X(488). (Floor van Lamoen, 4/26/98)
X(491) lies on these lines: 2,6 3,490 4,487 5,637 76,485 315,371 372,642 488,631
X(491) = isotomic conjugate of X(486)
X(491) = anticomplement of X(615)
X(491) = X(264)-Ceva conjugate of X(492)
X(491) = cevapoint of X(2) and X(487)
X(492)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 + csc A sin B sin C) + cos B cos CBarycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(492) is a pole associated with squares that circumscribe ABC. For details and reference, see X(488). (Floor van Lamoen, 4/27/98)
X(492) lies on these lines: 2,6 3,489 4,488 5,638 76,486 315,372 371,641 487,631
X(492) = isotomic conjugate of X(485)
X(492) = anticomplement of X(590)
X(492) = X(264)-Ceva conjugate of X(491)
X(492) = cevapoint of X(2) and X(488)
X(493)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sin A + sin B sin C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(493) is a homothetic center of triangles associated with squares that circumscribe ABC. For details, see Vierkanten in een driehoek: 4. Ingeschreven vierkanten (Floor van Lamoen, 4/27/98)
X(493) lies on these lines: 25,371 39,494
X(493) = X(394)-cross conjugate of X(494)
X(494)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sin A - sin B sin C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(494) is a homothetic center of triangles associated with squares that circumscribe ABC. For details and reference, see X(393). (Floor van Lamoen, 4/27/98)
X(494) lies on these lines: 25,372 39,493
X(494) = X(394)-cross conjugate of X(493)
X(495) = JOHNSON MIDPOINT
Trilinears 2 + cos(B - C) : 2 + cos(C - A) : 2 + cos(A - B)Barycentrics (sin A)[2 + cos(B - C)] : (sin B)[2 + cos(C - A)] : (sin C)[2 + cos(A - B)]
X(495) is the midpoint of segments C1-to-P1, C2-to-P2, C3-to-P3 in the Johnson four-circle configuration.
Roger A. Johnson, Advanced Euclidean Geometry, Dover, New York, 1960, page 75.
Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(495) is the point R on page 5.
X(495) lies on these lines:
1,5 2,956 3,388 4,390 8,442 10,141 30,55 35,550 36,549 56,140 202,395 203,396 226,517 381,497 392,908 429,1068 529,993 612,1060
X(495) = complement of X(956)
X(496) = HARMONIC CONJUGATE OF X(495) WRT X(1) AND X(5)
Trilinears 2 - cos(B - C) : 2 - cos(C - A) : 2 - cos(A - B)Barycentrics (sin A)[2 - cos(B - C)] : (sin B)[2 - cos(C - A)] : (sin C)[2 - cos(A - B)]
Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(496) is the point R' on page 5.
X(496) lies on these lines: 1,5 2,1058 3,497 4,999 30,56 35,549 36,550 55,140 149,404 202,397 203,398 381,388 390,631 613,1069 614,1062 942,946
X(497) CROSSPOINT OF GERGONNE POINT AND NAGEL POINT
Trilinears 1 - cos B cos C : 1 - cos C cos A : 1 - cos A cos BBarycentrics (sin A)(1 - cos B cos C) : (sin B)(1 - cos C cos A) : (sin C)(1 - cos A cos B)
X(497) is the harmonic conjugate of X(388) with respect to X(1) and X(4)
Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(497) is the point C' on page 5.
X(497) lies on these lines:
1,4 2,11 3,496 7,354 8,210 20,56 29,1036 30,999 35,499 36,376 57,516 65,938 69,350 80,1000 212,238 329,518 381,495 452,958 614,1040 1057,1065
X(497) = isogonal conjugate of X(1037)
X(497) = crosspoint of X(I) and X(J) for these (I,J): (7,8), (29,314)
X(498)
Trilinears 1 + 2 sin B sin C : 1 + 2 sin C sin A : 1 + 2 sin A sin BBarycentrics (sin A)(1 + 2 sin B sin C) : (sin B)(1 + 2 sin C sin A) : (sin C)(1 + 2 sin A sin B)
X(498) and X(499) are harmonic conjugate points with respect to X(1) and X(2), in analogy with such pairs with respect to X(1), X(4) and with respect to X(1), X(5).
Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(498) is the point S on page 6.
X(498) lies on these lines: 1,2 3,12 4,35 5,55 9,920 36,388 37,91 46,226 47,171 56,140 141,611 191,329 255,750 345,1089
X(499)
Trilinears 1 - 2 sin B sin C : 1 - 2 sin C sin A : 1 - 2 sin A sin BBarycentrics (sin A)(1 - 2 sin B sin C) : (sin B)(1 - 2 sin C sin A) : (sin C)(1 - 2 sin A sin B)
X(499) is the harmonic conjugate of X(498) with respect to X(1) and X(2).
Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(498) is the point S' on page 6.
X(499) lies on these lines: 1,2 3,11 4,36 5,56 12,999 17,202 18,203 35,497 46,946 47,238 55,140 57,920 80,944 141,613 255,748 348,1111 484,962
X(500) = ORTHOCENTER OF THE INCENTRAL TRIANGLE
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = a(b2 +c2 - a2 + bc)[2abc + (b + c)(a2 - (b - c)2)]
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(500) lies on these lines: 1,30 3,6 651,943
X(500) = inverse of X(582) in the Brocard circle
X(500) = crosspoint of X(1) and X(35)
X(501) = MIQUEL ASSOCIATE OF INCENTER
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = a/[(b + c)u], u = u(a,b,c) = a3 - b3 - c3 + ba2 - ab2 + ca2 - ac2 - cb2 - bc2 - abc
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Miquel's theorem states that if A', B', C' are points (other than A, B, C) on sidelines BC, CA, AB, respectively, then the circles AB'C', BC'A', CA'B' meet at a point. Suppose P is a point and A' = P^BC, B' = P^CA, C' = P^AB; the point in which the three circles is the Miquel associate of P. (Paul Yiu, 7/6/99)
X(501) lies on this line: 1,229
X(501) = isogonal conjugate of X(502)
X(501) = X(267)-Ceva conjugate of X(58)
X(502) = CYCLOCEVIAN CONJUGATE OF X(1)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (b + c)u/a, u = u(a,b,c) = a3 - b3 - c3 + ba2 - ab2 + ca2 - ac2 - cb2 - bc2 - abc
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
If A', B', C' are points (other than A, B, C) on sidelines BC, CA, AB, respectively, then the circle passing through A', B', C' passes through through points A" on BC, B" on CA, C" on AB. Suppose P is a point and A' = P^BC, B' = P^CA, C' = P^AB. Then lines AA", BB", CC" concur in the cyclocevian conjugate of P, as defined in TCCT p.226. (Paul Yiu, 7/6/99)
X(502) lies on these lines: 1,2 261,319
X(502) = isogonal conjugate of X(501)
X(502) = cyclocevian conjugate of X(1)>BR>
X(502) = X(191)-cross conjugate of X(10)
Centers 503- 510,
173, 174, 258, and 351- 364 are associated with isoscelizers. A line L perpendicular
to the line that bisects angle A is an A-isoscelizer. Let E = L∩CA and F = L∩AB;
then |AE| = |AF|. Centers defined by Peter Yff as points X of concurrence of A-, B-, and C-
isoscelizers depend on these notations:
t(X,A) = |AE|, h(X,A) = A-altitude of AEF, v(X,A) = |EF|.
X(503)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec B/2 + sec C/2 - sec A/2Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
The isoscelizer equations ah(X,A) = bh(X,B) = ch(X,C) have solution X = X(503). (Peter Yff, 4/9/99)
X(503) lies on these lines: 1,167 164,361 173,844
X(503) = X(259)-Ceva conjugate of X(1)
X(504)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = b sin B/2 + c sin C/2 - a sin A/2Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
The isoscelizer equations [h(X,A)]/a = [h(X,B)]/b = [h(X,C)]/c have solution X = X(504). (Peter Yff, 4/9/99)
X(504) lies on this line: 164,173
X(505)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sin B/2 + sin C/2 - sin A/2)Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
The isoscelizer equations h(X,A)v(X,A) = h(X,B)v(X,B) = h(X,C)v(X,C) have solution X = X(505). (Peter Yff, 4/9/99)
X(505) lies on this line: 40, 164
X(505) = isogonal conjugate of X(164)
X(505) = X(266)-cross conjugate of X(1)
X(506)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A/2)-2/3Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
The isoscelizer equations
have solution X = X(506). (Peter Yff, 4/9/99)
X(507)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A/2)-1/2Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
The isoscelizer equations
have solution X = X(507). (Peter Yff, 4/9/99)
X(508)
Trilinears a-1/2sec(A/2) : b-1/2sec(B/2) : c-1/2sec(C/2)Barycentrics a1/2sec(A/2) : b1/2sec(B/2) : c1/2sec(C/2)
The isoscelizer equations
have solution X = X(508). (Peter Yff, 4/9/99)
X(509)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan A/2)1/2Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)
The isoscelizer equations
have solution X = X(509). (Peter Yff, 4/9/99)
X(510)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3/2 + c3/2 - a3/2Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
The isoscelizer equations
have solution X = X(510). (Peter Yff, 4/9/99)
Centers 511- 526
and 30 lie on the line at infinity.
Thus, each collection of collinearities comprises a family of parallel lines.
X(511) = ISOGONAL CONJUGATE OF X(98)
Trilinears cos(A + ω) : cos(B + ω) : cos(C + ω)Barycentrics sin A cos(A + ω) : cos B cos(B + ω) : cos C cos(C + ω)
As the isogonal conjugate of a point on the circumcircle, X(511) lies on the line at infinity.
X(511) lies on these lines:
1,256 2,51 3,6 4,69 5,141 20,185 22,184 23,110 24,1092 25,394 26,206 30,512 40,1045 55,611 56,613 66,68 67,265 74,691 98,385 107,450 111,352 114,325 125,858 140,143 155,159 171,181 186,249 287,401 298,1080 299,383 343,427 381,599 549,597
X(511) = isogonal conjugate of X(98)
X(511) = isotomic conjugate of X(290)
X(511) = cevapoint of X(385) and X(401)
X(511) = X(I)-cross conjugate of X(J) for these (I,J): (4,114), (290,2), (297,232)
X(511) = crosspoint of X(I) and X(J) for these (I,J): (2,290), (297,325)
X(511) = X(3)-Hirst inverse of X(6)
X(511) = X(I)-line conjugate of X(J) for these (I,J): (3,6), (30,523)
X(512) = ISOGONAL CONJUGATE OF X(99)
Trilinears a(b2 - c2) : b(c2 - a2) : c(a2 - b2)Barycentrics a2(b2 - c2) : b2(c2 - a2) : c2(a2 - b2)
X(512) is the point in which the line of the 1st and 2nd Brocard points meets the line at infinity.
X(512) lies on these lines: 1,875 4,879 30,511 32,878 39,881 74,842 99,805 110,249 111,843 187,237 316,850 660,1016 670,886
X(512) = isogonal conjugate of X(99)
X(512) = isotomic conjugate of X(670)
X(512) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,115), (66,125), (99,39), (110,6), (112,32)
X(512) = crosspoint of X(I) and X(J) for these (I,J): (4,112), (6,110), (83,99)
X(512) = X(112)-line conjugate of X(30)
X(513) = ISOGONAL CONJUGATE OF X(100)
Trilinears b - c : c - a : a - bBarycentrics ab - ac : bc - ba : ca - cb
As the isogonal conjugate of a point on the circumcircle, X(513) lies on the line at infinity.
X(513) lies on these lines: 1,764 6,1024 7,885 30,511 36,238 37,876 44,649 59,651 100,765 104,953 105,840 190,660 320,350 663,855 668,889 1052,1054
X(513) = X(244)-cross conjugate of X(1)
X(513) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,244), (4,11), (100,1), (101,354), (108,56), (109,65), (190,37)
X(513) = isogonal conjugate of X(100)
X(513) = isotomic conjugate of X(668)
X(513) = crosspoint of X(I) and X(J) for these (I,J): (1,100), (4,108), (58,109), (86,190)
X(513) = X(I)-line conjugate of X(J) for these (I,J): (30,518), (36,238)
X(514) = ISOGONAL CONJUGATE OF X(101)
Trilinears (b - c)/a : (c - a)/b : (a - b)/cBarycentrics b - c : c - a : a - b
As the isogonal conjugate of a point on the circumcircle, X(514) lies on the line at infinity.
X(514) lies on this line: 1,663 2,1022 10,764 30,511 101,664 109,929 190,1016 239,649 241,650 551,676 651,655 659,667 661,693
X(514) = isogonal conjugate of X(101)
X(514) = isotomic conjugate of X(190)
X(514) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,116), (7,11), (75,244), (100,142), (190,2)
X(514) = X(I)-cross conjugate of X(J) for these (I,J): (11,7), (244,75)
X(514) = crosspoint of X(2) and X(190)
X(515) = ISOGONAL CONJUGATE OF X(102)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)sec A - b sec B - c sec CBarycentrics af(a,b,c) : bf(b,c,a) cf(c,a,b)
As the isogonal conjugate of a point on the circumcircle, X(515) lies on the line at infinity.
X(515) lies on these lines: 1,4 3,10 8,20 29,947 30,511 36,80 55,1012 103,929 119,214 153,908 165,376 281,610 284,1065 381,551
X(515) = isogonal conjugate of X(102)
X(515) = X(4)-Ceva conjugate of X(117)
X(516) = ISOGONAL CONJUGATE OF X(103)
Trilinears 1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) : f(b,c,a) : f(c,a,b) = X(103)Barycentrics a/f(a,b,c) : b/f(b,c,a) c/f(c,a,b)
As the isogonal conjugate of a point on the circumcircle, X(516) lies on the line at infinity.
X(516) lies on these lines:
1,7 2,165 3,142 4,9 8,144 30,511 35,411 55,226 57,497 65,950 80,655 100,908 102,929 103,927 118,910 200,329 238,673 354,553 355,382 376,551 993,1012
X(516) = isogonal conjugate of X(103)
X(516) = X(4)-Ceva conjugate of X(118)
X(517) = ISOGONAL CONJUGATE OF X(104)
Trilinears -1 + cos B + cos C : -1 + cos C + cos A : -1 + cos A + cos BBarycentrics (sin A)(-1 + cos B + cos C) : (sin B)(-1 + cos C + cos A) : (sin C)(-1 + cos A + cos B)
As the isogonal conjugate of a point on the circumcircle, X(517) lies on the line at infinity.
X(517) lies on these lines:
1,3 2,392 4,8 5,10 6,998 7,1000 9,374 19,219 20,145 30,511 37,573 42,1064 63,956 78,945 100,953 101,910 104,901 119,908 169,220 210,381 226,495 238,1052 389,950 549,551 572,1100 580,595 582,602 938,1058 1042,1066
X(517) = isogonal conjugate of X(104)
X(517) = X(4)-Ceva conjugate of X(119)
X(517) = crosspoint of X(I) and X(J) for these (I,J): (1,80), (7,88)
X(518) = ISOGONAL CONJUGATE OF X(105)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ab + ac - b2 - c2Barycentrics af(a,b,c) : bf(b,c,a) cf(c,a,b)
As the isogonal conjugate of a point on the circumcircle, X(518) lies on the line at infinity.
X(518) lies on these lines:
1,6 2,210 7,8 10,141 11,908 30,511 38,42 43,982 55,63 56,78 57,200 59,765 144,145 209,306 239,335 244,899 329,497 551,597 583,1009 612,940 896,902 997,999
X(518) = isogonal conjugate of X(105)
X(518) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,120), (335,37)
X(518) = crosspoint of X(1) and X(291)
X(518) = X(1)-Hirst inverse of X(9)
X(518) = X(I)-line conjugate of X(J) for these (I,J): (1,6), (30,513)
X(519) = ISOGONAL CONJUGATE OF X(106)
Trilinears (2a - b - c)/a : (2b - c - a)/b : (2c - a - b)/cBarycentrics 2a - b - c : 2b - c - a : 2c - a - b
As the isogonal conjugate of a point on the circumcircle, X(519) lies on the line at infinity.
X(519) lies on these lines: 1,2 6,996 9,1000 30,511 36,100 40,376 55,956 58,1043 65,553 72,950 80,908 210,392 238,765 320,668 355,381 447,648 594,1100 672,1018 751,984
X(519) = isogonal conjugate of X(106)
X(519) = isotomic conjugate of X(903)
X(519) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,121), (80,10)
X(520) = ISOGONAL CONJUGATE OF X(107)
Trilinears (cos A)(sin 2B - sin 2C) : (cos B)(sin 2C - sin 2A) : (cos C)(sin 2A - sin 2B)Barycentrics (sin 2A)(sin 2B - sin 2C) : (sin 2B)(sin 2C - sin 2A) : (sin 2C)(sin 2A - sin 2B)
As the isogonal conjugate of a point on the circumcircle, X(520) lies on the line at infinity.
X(520) lies on these lines: 30,511 69,879 110,250 340,850 647,652
X(520) = isogonal conjugate of X(107)
X(520) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,122), (68,125), (110,3)
X(520) = crosspoint of X(3) and X(110)
X(521) = ISOGONAL CONJUGATE OF X(108)
Trilinears (sec B - sec C)(csc A) : (sec C - sec A)(csc B) : (sec A - sec B)(csc C)Barycentrics sec B - sec C : sec C - sec A : sec A - sec B
As the isogonal conjugate of a point on the circumcircle, X(521) lies on the line at infinity.
X(521) lies on these lines: 30,511 59,100 650,1021 656,810
X(521) = isogonal conjugate of X(108)
X(521) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,123), (100,3)
X(521) = crosspoint of X(8) and X(100)
X(522) = ISOGONAL CONJUGATE OF X(109)
Trilinears (cos B - cos C)(csc A) : (cos C - cos A)(csc B) : (cos A - cos B)(csc C)Barycentrics cos B - cos C : cos C - cos A : cos A - cos B
As the isogonal conjugate of a point on the circumcircle, X(522) lies on the line at infinity.
X(522) lies on this line: 9,657 30,511 100,655 101,929 190,666 240,656 243,652
X(522) = isogonal conjugate of X(109)
X(522) = isotomic conjugate of X(664)
X(522) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,124), (8,11), (100,10), (190,9)
X(522) = X(11)-cross conjugate of X(8)
X(522) = crosspoint of X(I) and X(J) for these (I,J): (21,100), (75,190)
X(523) = ISOGONAL CONJUGATE OF X(110)
Trilinears sin(B - C) : sin(C - A) : sin(A - B)Barycentrics b2 - c2 : c2 - a2 : a2 - b2
As the isogonal conjugate of a point on the circumcircle, X(523) lies on the line at infinity.
X(523) lies on these lines: 6,879 11,1090 23,385 30,511 69,655 74,477 75,876 98,842 99,691 110,476 112,935 141,882 230,231 250,648 325,684
X(523) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,11), (2,115), (4,125), (99,2), (100,442), (107,4), (108,429), (108,429), (110,5), (112,427), (254,136), (476,30)
X(523) = isogonal conjugate of X(110)
X(523) = isotomic conjugate of X(99)
X(523) = cevapoint of X(2) and X(148)
X(523) = X(I)-cross conjugate of X(J) for these (I,J): (115,2), (125,4)
X(523) = crosspoint of X(I) and X(J) for these (I,J): (2,99), (4,107), (54,110), (112,251)
X(523) = X(30)-line conjugate of X(511)
X(524) = ISOGONAL CONJUGATE OF X(111)
Trilinears (2a2 - b2 - c2)/a : (2b2 - c2 - a2)/b : (2c2 - a2 - b2)/cBarycentrics 2a2 - b2 - c2 : 2b2 - c2 - a2 : 2c2 - a2 - b2
As the isogonal conjugate of a point on the circumcircle, X(524) lies on the line at infinity.
X(524) lies on these lines: 2,6 5,576 30,511 53,317 67,858 76,598 99,843 140,575 182,549 239,320 297,340 316,594 319,594 397,633 398,634
X(524) = isogonal conjugate of X(111)
X(524) = isotomic conjugate of X(671)
X(524) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,126), (67,141)
X(524) = X(187)-cross conjugate of X(468)
X(524) = X(I)-line conjugate of X(J) for these (I,J): (2,6), (30,512)
X(525) = ISOGONAL CONJUGATE OF X(112)
Trilinears (b cos B - c cos C)/a : (c cos C - a cos A)/b : (a cos A - b cos B)/cBarycentrics b cos B - c cos C : c cos C - a cos A : a cos A - b cos B
As the isogonal conjugate of a point on the circumcircle, X(525) lies on the line at infinity.
X(525) lies on these lines: 3,878 30,511 99,249 110,935 297,850 323,401 441,647
X(525) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,127), (69,125), (76,339), (99,3), (110,141), (190,440), (253, 122)
X(525) = isogonal conjugate of X(112)
X(525) = isotomic conjugate of X(648)
X(525) = X(I)-cross conjugate of X(J) for these (I,J): (115,68), (122,253), (125,69)
X(525) = crosspoint of X(76) and X(99)
X(526) = ISOGONAL CONJUGATE OF X(476)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 + 2 cos 2A)sin(B - C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
As the isogonal conjugate of a point on the circumcircle, X(526) lies on the line at infinity.
X(526) = isogonal conjugate of X(476)
X(526) lies on these lines: 30,511 67,879 110,351
X(526) = crosspoint of X(74) and X(110)
Centers 527- 565
were added to ETC on 1/1/01.
X(527) = DIRECTION OF VECTOR AX + BX + CX, where X = X(7)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = 1/[a(b + c - a)], y = x(b,c,a), z = x(c,a,b)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(527) lies on the line at infinity.
X(527) lies on these lines: 2,7 30,511 44,1086 190,320 551,993 666,673
X(528) = DIRECTION OF VECTOR AX + BX + CX, where X = X(11)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = 1 - cos(B-C), y = x(B,C,A), z = x(C,A,B)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(528) lies on the line at infinity.
X(528) lies on these lines: 1,1086 2,11 7,664 8,190 9,80 30,511 104,376 119,381 142,214
X(528) = isogonal conjugate of X(840)
X(529) = DIRECTION OF VECTOR AX + BX + CX, where X = X(12)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = 1 + cos(B-C), y = x(B,C,A), z = x(C,A,B)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(529) lies on the line at infinity.
X(529) lies on these lines: 2,12 30,511 495,993 1001,1056
X(530) = DIRECTION OF VECTOR AX + BX + CX, where X = X(13)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = csc(A + π/3), y = x(B,C,A), z = x(C,A,B)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(530) lies on the line at infinity.
X(530) lies on these lines: 2,13 14,671 30,511 99,299 115,395 187,396 298,316
X(531) = DIRECTION OF VECTOR AX + BX + CX, where X = X(14)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = csc(A - π/3), y = x(B,C,A), z = x(C,A,B)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(531) lies on the line at infinity.
X(531) lies on these lines: 2,14 13,671 30,511 99,298 115,396 187,395 299,316
X(532) = DIRECTION OF VECTOR AX + BX + CX, where X = X(17)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = csc(A + π/6), y = x(B,C,A), z = x(C,A,B)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(532) lies on the line at infinity.
X(532) lies on these lines: 2,17 13,298 14,622 15,616 16,299 30,511 395,624 396,618 397,635
X(533) = DIRECTION OF VECTOR AX + BX + CX, where X = X(18)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = csc(A - π/6), y = x(B,C,A), z = x(C,A,B)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(533) lies on the line at infinity.
X(533) lies on these lines: 2,18 13,621 14,299 15,298 16,617 30,511 395,619 396,623 398,636
X(534) = DIRECTION OF VECTOR AX + BX + CX, where X = X(19)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = tan A, y = x(B,C,A), z = x(C,A,B)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(534) lies on the line at infinity.
X(534) lies on these lines: 2,19 30,511
X(535) = DIRECTION OF VECTOR AX + BX + CX, where X = X(36)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = 1 - 2 cos A, y = x(B,C,A), z = x(C,A,B)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(535) lies on the line at infinity.
X(535) lies on these lines: 2,36 30,511 63,484 214,908 226,551
X(536) = DIRECTION OF VECTOR AX + BX + CX, where X = X(37)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = b + c, y = x(b,c,a), z = x(c,a,b)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(536) lies on the line at infinity.
X(536) lies on these lines: 2,37 30,511 44,190 335,903 894,1100
X(536) = isogonal conjugate of X(739)
X(537) = DIRECTION OF VECTOR AX + BX + CX, where X = X(38)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = b2 + c2, y = x(b,c,a), z = x(c,a,b)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(537) lies on the line at infinity.
X(537) lies on these lines: 1,190 2,38 10,1086 30,511 37,551 75,668
X(538) = DIRECTION OF VECTOR AX + BX + CX, where X = X(39)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = a(b2 + c2), y = x(b,c,a), z = x(c,a,b)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(538) lies on the line at infinity.
X(538) lies on these lines: 2,39 30,511 32,1003 99,187 115,325 148,316 183,574 230,620 350,1015
X(538) = isogonal conjugate of X(729)
X(539) = DIRECTION OF VECTOR AX + BX + CX, where X = X(54)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = 1/cos(B-C), y = x(B,C,A), z = x(C,A,B)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(539) lies on the line at infinity.
X(539) lies on these lines: 2,54 30,511 155,195
X(540) = DIRECTION OF VECTOR AX + BX + CX, where X = X(58)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = a/(b + c), y = x(b,c,a), z = x(c,a,b)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(540) lies on the line at infinity.
X(540) lies on these lines: 2,58 30,511 340,447
X(541) = DIRECTION OF VECTOR AX + BX + CX, where X = X(74)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = 1/(cos A - 2 cos B cos C), y = x(B,C,A), z = x(C,A,B)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(541) lies on the line at infinity.
X(541) lies on these lines: 2,74 30,511 110,376 125,381 394,399
X(541) = isogonal conjugate of X(841)
X(542) = DIRECTION OF VECTOR AX + BX + CX, where X = X(98)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = sec(A + ω), y = x(B,C,A), z = x(C,A,B)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(542) lies on the line at infinity.
X(542) lies on these lines: 2,98 3,67 4,576 5,575 6,13 30,511 69,74 141,549 146,148
X(542) = isogonal conjugate of X(842)
X(543) = DIRECTION OF VECTOR AX + BX + CX, where X = X(99)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = bc/(b2 - c2), y = x(b,c,a), z = x(c,a,b)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(543) lies on the line at infinity.
X(543) lies on these lines: 2,99 30,511 98,376 114,381
X(543) = isogonal conjugate of X(843)
X(544) = DIRECTION OF VECTOR AX + BX + CX, where X = X(101)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = a/(b - c), y = x(b,c,a), z = x(c,a,b)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(544) lies on the line at infinity.
X(544) lies on these lines: 2,101 30,511 63,1018 103,376 118,381
X(545) = DIRECTION OF VECTOR AX + BX + CX, where X = X(190)
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = bc/(b - c), y = x(b,c,a), z = x(c,a,b)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(545) lies on the line at infinity.
X(545) lies on these lines: 2,45 30,511
X(546) = NINE-POINT CENTER OF X(4)-EULER MIDWAY TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = 3 cos(B - C) - 2 cos A
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(546) lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)
X(546) lies on these lines: 2,3 13,398 14,397 113,137 156,578 946,952
X(546) = inverse of X(382) in the orthocentroidal circle
X(546) = complement of X(550)
X(547) = CENTROID OF X(5)-EULER MIDWAY TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = 5 cos(B - C) + 2 cos A
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(547) lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)
X(547) lies on these lines: 2,3 551,952
X(547) = complement of X(549)
X(548) = DE LONGCHAMPS POINT OF X(5)-EULER MIDWAY TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = - cos(B - C) + 6 cos A
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(548) lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)
X(549) = ORTHOCENTER OF X(2)-EULER MIDWAY TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = cos(B - C) + 4 cos A
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(549) lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)
X(549) lies on these lines: 2,3 15,395 16,396 35,496 36,495 141,542 182,524 230,574 302,617 303,616 511,597 517,551
X(549) = complement of X(381)
X(549) = anticomplement of X(547)
X(550) = ORTHOCENTER OF X(20)-EULER MIDWAY TRIANGLE
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = - cos(B - C) + 4 cos A
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(550) lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)
X(550) lies on these lines: 2,3 15,397 16,398 35,495 36,496 40,952 74,930 156,1092 165,355
X(550) = complement of X(382) X(550) = anticomplement of X(546)
X(551) = INCENTER OF X(1)-EULER MIDWAY TRIANGLE
Trilinears (4a + b + c)/a : (4b + c + a)/b : (4c + a + b)/cBarycentrics 4a + b + c : 4b + c + a : 4c + a + b
X(551) lies on these lines: 1,2 30,946 37,537 56,553 86,99 142,214 226,535 354,392 376,516 381,515 514,676 517,549 518,597 527,993 547,952
(Antreas Hatzipolakis, 1/24/00, Hyacinthos #223)
X(552)
Trilinears 1/[a(b + c - a)(b + c)2] : 1/[b(c +a - b)(c + a)2] : 1/[c(a + b - c)(a + b)2]Barycentrics 1/[(b+c-a)(b+c)2] : 1/[(c+a-b)(c+a)2] : 1/[(a+b-c)(a+b)2]
X(552) lies on this line: 261,873
(Floor van Lamoen, 1/30/00, Hyacinthos #255)
X(553)
Trilinears bc(2a + b + c)/(b + c - a) : ca(2b + c + a)/(c + a - b) : ab/(2c + a + b)/(a + b - c)Barycentrics (2a + b + c)/(b + c - a) : (2b + c + a)/(c + a - b) : /(2c + a + b)/(a + b - c)
X(553) lies on these lines: 1,376 2,7 30,942 56,551 65,519 354,516
(Floor van Lamoen, 1/30/00, Hyacinthos #257)
X(554)
Trilinears sec(A/2) csc(A/2 + π/3) : sec(B/2) csc(B/2 + π/3) : sec(C/2) csc(C/2 + π/3)Barycentrics sin A sec(A/2) csc(A/2 + π/3) : sin B sec(B/2) csc(B/2 + π/3) : sin C sec(C/2) csc(C/2 + π/3)
X(554) lies on these lines: 1,30 7,1082 14,226 75,299
Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438; in (9) on page 435, take Y = X(13).
X(555)
Trilinears sec3(A/2) : sec3(B/2) : sec3(C/2)Barycentrics sin A sec3(A/2) : sin B sec3(B/2) : sin C sec3(C/2)
X(555) lies on these lines: 7,177 234,1088
Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438; in (9) on page 435, take Y = X(7).
X(556)
Trilinears csc A csc A/2 : csc B csc B/2 : csc C csc C/2Barycentrics csc A/2 : csc B/2 : csc C/2
X(556) lies on these lines: 8,177 74,234
X(556) = isotomic conjugate of X(174)
Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438; in (9) on page 435, take Y = X(8).
X(557)
Trilinears sec A/2 cot A/4 : sec B/2 cot B/4 : sec C/2 cot C/4Barycentrics cos A/2 cot A/4 : cos B/2 cot B/4 : cos C/2 cot C/4
X(557) lies on this line: 2,178
Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438; in (9) on page 435, take Y = X(9).
X(558)
Trilinears sec A/2 tan A/4 : sec B/2 tan B/4 : sec C/2 tan C/4Barycentrics sin2(A/4) : sin2(B/4) : sin2(C/4)
X(558) lies on this line: 2,178
Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438; in (9) on page 435, take Y = X(57).
X(559)
Trilinears (sec A/2) sin(A/2 + π/3) : (sec B/2) sin(B/2 + π/3) : (sec C/2) sin(C/2 + π/3)Barycentrics (sin A/2) sin(A/2 + π/3) : (sin B/2) sin(B/2 + π/3) : (sin C/2) sin(C/2 + π/3)
X(559) lies on these lines: 1,3 14,226 299,319
Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438; in (9) on page 435, take Y = X(15).
X(560) 4th POWER POINT
Trilinears a4 : b4 : c4Barycentrics a5 : b5 : c5
X(560) lies on these lines: 1,82 31,48 41,872 42,584 100,697 101,713 110,715 717,825 719,827
X(560) = isogonal conjugate of X(561)
X(561) ISOGONAL CONJUGATE OF 4th POWER POINT
Trilinears a - 4 : b - 4 : c - 4Barycentrics a - 3 : b - 3 : c - 3
X(561) lies on these lines: 1,718 2,716 6,720 31,722 32,724 38,75 63,799 76,321 92,304 313,696
X(561) = isogonal conjugate of X(560)
X(561) = isotomic conjugate of X(31)
X(562)
Trilinears csc A tan 3A : csc B tan 3B : csc C tan 3CBarycentrics tan 3A : tan 3B : tan 3C
X(562) lies on this line: 4,93
Clark Kimberling and Cyril Parry, "Products, Square Roots, and Layers in Triangle Geometry," Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.
X(563)
Trilinears sin 4A : sin 4B : sin 4CBarycentrics sin A sin 4A : sin B sin 4B : sin C sin 4C
X(563) lies on these lines: 19,163 48,255
Clark Kimberling and Cyril Parry, "Products, Square Roots, and Layers in Triangle Geometry," Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.
X(564)
Trilinears cos(2B - 2C) : cos(2C - 2A) : cos(2A - 2B)Barycentrics sin A cos(2B - 2C) : sin B cos(2C - 2A) : sin C cos(2A - 2B)
X(564) lies on these lines: 1,1048 47,91
Clark Kimberling and Cyril Parry, "Products, Square Roots, and Layers in Triangle Geometry," Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.
X(565)
Trilinears cos(3B - 3C) : cos(3C - 3A) : cos(3A - 3B)Barycentrics sin A cos(3B - 3C) : sin B cos(3C - 3A) : sin C cos(3A - 3B)
X(565) lies on these lines: 49,93 143,324
Clark Kimberling and Cyril Parry, "Products, Square Roots, and Layers in Triangle Geometry," Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.
Centers 566- 584
are on the Brocard axis, L(3,6). Each is the center X of a circle
meeting the sides of triangle ABC with three equal angles at X.
Let AB, AC, BC, BA, CA, CB denote the meeting-points; e.g., AB and CB
are on side CA. The equal angles are given by
D = angle(ABXAC) = angle(BCXBA) = angle(CAXCB)
Then trilinears for X are given by
X = sin A + cot D/2 cos A : sin B + cot D/2 cos B : sin C + cot D/2 cos C.
Definitions: Y is the orthogonal of X if D(X) + D(Y) = π/2;
Y is the harmonic of X if X and Y are harmonic conjugates with respect to X(3) and X(6);
Y is the orthoharmonic if Y is the harmonic of the orthogonal of X.
For a figure and details, see
Edward Brisse, "Some properties of points on the axis of Brocard."
X(566) = HARMONIC OF X(50)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = 4*area/(a2 + b2 + c2 - 6r2), where r = abc/(4*area)
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(566) lies on these lines: 2,94 3,6
X(566) = inverse of X(50) in the Brocard circle
Edward Brisse, as cited above X(566), page 7.
X(567) = ORTHOGONAL OF X(50)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (a2 + b2 + c2 - 6r2)/(4*area), where r = abc/(4*area)
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(567) lies on these lines: 3,6 5,49 184,381
X(567) = inverse of X(568) in the Brocard circle
Edward Brisse, as cited above X(566), page 7.
X(568) = ORTHOHARMONIC OF X(50)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (6r2 - a2 - b2 - c2)/(4*area), where r = abc/(4*area)
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(568) lies on these lines: 3,6 4,94 24,49 51,381 68,973 185,382
X(568) = inverse of X(567) in the Brocard circle
Edward Brisse, as cited above X(566), page 7.
X(569) = HARMONIC OF X(52)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (2e3 + e2 - e1)/[64*(area)3], where
e1 = a6 + b6 + c6
e2 = a4(b2 + c2) + b4(c2 + a2) + c4(a2 + b2)
e3 = a2b2c2
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(569) lies on these lines: 2,54 3,6 5,156 26,51 140,343
X(569) = inverse of X(52) in the Brocard circle
Edward Brisse, as cited above X(566), page 7.
X(570) = ORTHOGONAL OF X(52)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = [64*(area)3]/(2e3 + e2 - e1), where
e1, e2, e3 are as for X(569)
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(570) lies on these lines: 2,311 3,6 53,232 115,128 140,231 157,184
X(570) = inverse of X(571) in the Brocard circle
X(570) = complement of X(311)
Edward Brisse, as cited above X(566), page 7.
X(571) = ORTHOHARMONIC OF X(52)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = [64*(area)3]/(e1 - 2e3 - e2), where
e1, e2, e3 are as for X(569)
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(571) lies on these lines: 3,6 4,96 66,248 112,393 160,184 206,237 230,427 608,913
X(571) = inverse of X(570) in the Brocard circle
Edward Brisse, as cited above X(566), page 7.
X(572) = ORTHOGONAL OF X(58)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (a + b + c)2/(4*area)
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(572) lies on these lines: 1,604 3,6 9,48 51,199 54,71 103,825 165,1051 169,610 184,1011 219,947 261,662 517,1100 594,952 631,966
X(572) = inverse of X(573) in the Brocard circle
Edward Brisse, as cited above X(566), page 7.
X(573) = ORTHOHARMONIC OF X(58)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = - (a + b + c)2/(4*area)
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(573) lies on these lines: 1,941 3,6 4,9 20,391 36,604 37,517 43,165 51,1011 55,181 101,102 109,478 184,199 256,981 346,1018 347,1020
X(573) = inverse of X(572) in the Brocard circle
Edward Brisse, as cited above X(566), page 7.
X(574) = HARMONIC OF X(187)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = 12*area/(a2 + b2 + c2)
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(574) lies on these lines: 2,99 3,6 55,1015 110,353 183,538 230,549 232,378 805,843
X(574) = isogonal conjugate of X(598)
X(574) = inverse of X(187) in the Brocard circle
Edward Brisse, as cited above X(566), page 7.
X(575) = ORTHOGONAL OF X(187)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (a2 + b2 + c2)/(12*area)
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(575) lies on these lines: 3,6 4,598 5,542 23,51 54,895 110,373 140,524 141,629
X(575) = inverse of X(576) in the Brocard circle
Edward Brisse, as cited above X(566), page 7.
X(576) = ORTHOHARMONIC OF X(187)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = - (a2 + b2 + c2)/(12*area)
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(576) lies on these lines: 3,6 4,542 5,524 23,184 140,597 262,385
X(576) = inverse of X(575) in the Brocard circle
Edward Brisse, as cited above X(566), page 7.
X(577) = HARMONIC OF X(216)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = - 8*(area)3/(a2b2c2 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(577) lies on these lines: 2,95 3,6 20,393 22,232 30,53 48,603 69,248 112,376 141,441 160,206 172,1038 184,418 198,478 219,906 220,268 264,401 395,466 396,465
X(577) = inverse of X(216) in the Brocard circle
X(577) = complement of X(317)
Edward Brisse, as cited above X(566), page 7.
X(578) = ORTHOHARMONIC OF X(216)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (a2b2c2 cos A cos B cos C)/[8*(area)3]
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(578) lies on these lines: 2,1092 3,6 4,54 24,51 49,381 156,546 185,378 436,1093
X(578) = inverse of X(389) in the Brocard circle
Edward Brisse, as cited above X(566), page 7.
X(579) = HARMONIC OF X(284)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = 4*area*(a + b + c)/(e1 - e2 - 2abc), where
e1 = a3 + b3 + c3
e2 = a2(b + c) + b2(c + a) + c2(a + b)
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(579) lies on these lines: 1,71 2,7 3,6 19,46 36,48 37,942 40,387 56,219 109,608 165,380 198,218 443,966 474,965 517,1108
X(579) = inverse of X(284) in the Brocard circle
Edward Brisse, as cited above X(566), page 7.
X(580) = ORTHOGONAL OF X(284)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (e1 - e2 - 2abc)/[4*area*(a + b + c)], where e1, e2 are as for X(579)
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(580) lies on these lines: 1,201 2,283 3,6 31,40 34,46 36,54 57,255 162,412 165,601 223,603 238,946 517,595
X(580) = inverse of X(581) in the Brocard circle
Edward Brisse, as cited above X(566), page 7.
X(581) = ORTHOHARMONIC OF X(284)
Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (2abc - e1 + e2)/[4*area*(a + b + c)], where e1, e2 are as for X(579)
Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(581) lies on these lines: 1,4 3,6 35,47 40,42 81,411 84,941 222,1035 936,966&