These are points that have π/3 or π/6 in them. These turn into the square root of three which gives two versions.

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

Trilinears csc(A + π/3) : csc(B + π/3) : csc(C + π/3)

= sec(A - π/6) : sec(B - π/6) : sec(C - π/6)

X(13) lies on these lines:

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

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

X(13) = reflection of X(I) in X(J) for these (I,J): (14,115), (15,396), (99,619), (298,623), (616,618)

X(13) = isogonal conjugate of X(15)

X(13) = isotomic conjugate of X(298)

X(13) = inverse-in-orthocentroidal-circle of X(14)

X(13) = complement of X(616)

X(13) = anticomplement of X(618)

X(14) = 2nd ISOGONIC CENTER

Trilinears csc(A - π/3) : csc(B - π/3) : csc(C - π/3)

= sec(A + π/6) : sec(B + π/6) : sec(C + π/6)

X(14) lies on these lines:

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

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

X(14) = reflection of X(I) in X(J) for these (I,J): (13,115), (16,395), (99,618), (299,624), (617,619)

X(14) = isogonal conjugate of X(16)

X(14) = isotomic conjugate of X(299)

X(14) = inverse-in-orthocentroidal-circle of X(13)

X(14) = complement of X(617)

X(14) = anticomplement of X(619)

X(14) = cevapoint of X(16) and X(61)

X(14) = X(I)-cross conjugate of X(J) for these (I,J): (16,17), (30,13), (395,2)

------------------------------------------------------------------------

X(15) = 1st ISODYNAMIC POINT

Trilinears sin(A + π/3) : sin(B + π/3) : sin(C + π/3)

= cos(A - π/6) : cos(B - π/6) : cos(C - π/6)

X(15) lies on these lines:

1,1251 2,14 3,6 4,17 13,30 18,140 35,1250 36,202 55,203 298,533 303,316 395,549 397,550 532,616 628,636

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

X(15) = reflection of X(I) in X(J) for these (I,J): (13,396), (16,187), (298,618), (316,624), (621,623)

X(15) = isogonal conjugate of X(13)

X(15) = isotomic conjugate of X(300)

X(15) = inverse-in-circumcircle of X(16)

X(15) = inverse-in-Brocard-circle of X(16)

X(15) = complement of X(621)

X(15) = anticomplement of X(623)

X(15) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,202), (13,62), (74,16)

X(15) = crosspoint of X(I) and X(J) for these (I,J): (13,18), (298,470)

X(15) = crosssum of X(I) and X(J) for these (I,J): (15,62), (532,619)

X(15) = crossdifference of any two points on line X(395)X(523)

X(15) = X(6)-Hirst inverse of X(16)

------------------------------------------------------------------------

X(16) = 2nd ISODYNAMIC POINT

Trilinears sin(A - π/3) : sin(B - π/3) : sin(C - π/3)

= cos(A + π/6) : cos(B + π/6) : cos(C + π/6)

X(16) lies on these lines:

1,1250 2,13 3,6 4,18 14,30 17,140 36,203 55,202 299,532 302,316 358,1135 396,549 398,550 533,617 627,635

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

X(16) = reflection of X(I) in X(J) for these (I,J): (14,395), (15,187), (299,619), (316,623), (622,624)

X(16) = isogonal conjugate of X(14)

X(16) = isotomic conjugate of X(301)

X(16) = inverse-in-circumcircle of X(15)

X(16) = inverse-in-Brocard-circle of X(15)

X(16) = complement of X(622)

X(16) = anticomplement of X(624)

X(16) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,203), (14,61), (74,15)

X(16) = crosspoint of X(I) and X(J) for these (I,J): (14,17), (299,471)

X(16) = crosssum of X(I) and X(J) for these (I,J): (16,61), (533,618)

X(16) = crossdifference of any two points on line X(396)X(523)

X(16) = X(6)-Hirst inverse of X(15)

------------------------------------------------------------------------

X(17) = 1st NAPOLEON POINT

Trilinears csc(A + π/6) : csc(B + π/6) : csc(C + π/6)

= sec(A - π/3) : sec(B - π/3) : sec(C - π/3)

X(17) lies on these lines:

2,62 3,13 4,15 5,14 6,18 12,203 16,140 76,303 83,624 202,499 275,471 299,635 623,633

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

X(17) = reflection of X(627) in X(629)

X(17) = isogonal conjugate of X(61)

X(17) = isotomic conjugate of X(302)

X(17) = complement of X(627)

X(17) = anticomplement of X(629)

X(17) = X(I)-cross conjugate of X(J) for these (I,J): (16,14), (140,18), (397,4)

------------------------------------------------------------------------

X(18) = 2nd NAPOLEON POINT

Trilinears csc(A - π/6) : csc(B - π/6) : csc(C - π/6)

= sec(A + π/3) : sec(B + π/3) : sec(C + π/3)

X(18) lies on these lines:

2,61 3,14 4,16 5,13 6,17 12,202 15,140 76,302 83,623 203,499 275,470 298,636 624,634

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

X(18) = reflection of X(628) in X(630)

X(18) = isogonal conjugate of X(62)

X(18) = isotomic conjugate of X(303)

X(18) = complement of X(628)

X(18) = anticomplement of X(630)

X(18) = X(I)-cross conjugate of X(J) for these (I,J): (15,13), (140,17), (398,4)

X(61) = ISOGONAL CONJUGATE OF X(17)

Trilinears sin(A + π/6) : sin(B + π/6) : sin(C + π/6)

= cos(A - π/3) : cos(B - π/3) : cos(C - π/3)

X(61) lies on these lines:

1,203 2,18 3,6 4,13 5,14 30,397 56,202 140,395 299,636 302,629 618,627

X(61) is the {X(3),X(6)}-harmonic conjugate of X(62). For a list of other harmonic conjugates of X(61), click More at the top of this page.

X(61) = reflection of X(633) in X(635)

X(61) = isogonal conjugate of X(17)

X(61) = inverse-in-Brocard-circle of X(62)

X(61) = complement of X(633)

X(61) = anticomplement of X(635)

X(61) = eigencenter of cevian triangle of X(14)

X(61) = eigencenter of anticevian triangle of X(16)

X(61) = X(14)-Ceva conjugate of X(16)

X(61) = crosspoint of X(302) and X(473)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X = X(61); then W = X(397)X(61).

------------------------------------------------------------------------

X(62) = ISOGONAL CONJUGATE OF X(18)

Trilinears sin(A - π/6) : sin(B - π/6) : sin(C - π/6)

= cos(A + π/3) : cos(B + π/3) : cos(C + π/3)

X(62) lies on these lines:

1,202 2,17 3,6 4,14 5,13 30,398 56,203 140,396 298,635 303,630 619,628

X(62) is the {X(3),X(6)}-harmonic conjugate of X(61). For a list of other harmonic conjugates of X(62), click More at the top of this page.

X(62) = reflection of X(634) in X(636)

X(62) = isogonal conjugate of X(18)

X(62) = inverse-in-Brocard-circle of X(61)

X(62) = complement of X(634)

X(62) = anticomplement of X(636)

X(62) = eigencenter of cevian triangle of X(13)

X(62) = eigencenter of anticevian triangle of X(15)

X(62) = X(13)-Ceva conjugate of X(15)

X(62) = crosspoint of X(303) and X(472)

X(202) = X(1)-CEVA CONJUGATE OF X(15)

Trilinears u(v + w - u),

u = u(A,B,C) = sin(A + π/3), v = u(B,C,A), w = u(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)

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(298) ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER

Trilinears csc2A sin(A + π/3) : csc2B sin(B + π/3) : csc2C sin(C + π/3)

X(298) lies on these lines:

2,6 3,617 5,634 13,532 14,76 15,533 18,636 99,531 140,628 264,472 316,530 317,473 319,1082 340,470 381,622 511,1080

X(298) = midpoint of X(616) and X(621)

X(298) = reflection of X(I) in X(J) for these (I,J): (13,623), (15,618), (299,325), (385,395)

X(298) = 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)

X(299) lies on these lines:

2,6 3,616 5,633 13,76 14,533 16,532 17,635 30,617 75,554 99,530 140,627 264,473 316,531 317,472 319,559 340,471 381,621 383,511

X(299) = midpoint of X(617) and X(622)

X(299) = reflection of X(I) in X(J) for these (I,J): (14,624), (16,619), (298,325), (385,396)

X(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)

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)

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)

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)

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(383) = EULER LINE INTERCEPT OF LINE X(14)X(98)

Trilinears csc(B - C) [sin 2B cos(C - É÷) sin(C + π/3) - sin 2C cos(B - É÷) sin(B + π/3)]

X(383) lies on these lines: 2,3 13,262 14,98 183,621 299,511 325,622

X(383) = reflection of X(1080) in X(1513)

X(383) = inverse-in-orthocentroidal-circle of X(1080)

X(395) = MIDPOINT OF X(14) AND X(16)

Trilinears cos(B - C) + 2 cos(A + π/3)

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) is the {X(2),X(6)}-harmonic conjugate of X(396). For a list of other harmonic conjugates of X(395), click More at the top of this page.

X(395) = midpoint of X(I) and X(J) for these (I,J): (14,16), (298,385)

X(395) = reflection of X(396) in X(230)

X(395) = complement of X(299)

X(395) = crosspoint of X(2) and X(14)

X(395) = crosssum of X(6) and X(16)

X(395) = crossdifference of any two points on line X(15)X(512)

------------------------------------------------------------------------

X(396) = MIDPOINT OF X(13) AND X(15)

Trilinears cos(B - C) + 2 cos(A - π/3)

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) is the {X(2),X(6)}-harmonic conjugate of X(395). For a list of other harmonic conjugates of X(396), click More at the top of this page.

X(396) = midpoint of X(I) and X(J) for these (I,J): (13,15), (299,385)

X(396) = reflection of X(395) in X(230)

X(396) = isogonal conjugate of X(2981)

X(396) = anticomplement of X(298)

X(396) = crosspoint of X(2) and X(13)

X(396) = crosssum of X(6) and X(15)

X(396) = crossdifference of any two points on line X(16)X(512)

------------------------------------------------------------------------

X(397) CROSSPOINT OF X(4) AND X(17)

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)

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) is the {X(4),X(6)}-harmonic conjugate of X(398). For a list of other harmonic conjugates of X(397), click More at the top of this page.

X(397) = crosspoint of X(4) and X(17)

X(397) = crosssum of X(3) and X(61)

------------------------------------------------------------------------

X(398) CROSSPOINT OF X(4) AND X(18)

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)

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) is the {X(4),X(6)}-harmonic conjugate of X(397). For a list of other harmonic conjugates of X(398), click More at the top of this page.

> X(398) = crosspoint of X(4) and X(18)

X(398) = crosssum of X(3) and X(62)

X(462) = EULER LINE INTERCEPT OF LINE X(51)X(397)

Trilinears u tan A : v tan B : w tan C, where

u : 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) = EULER LINE INTERCEPT OF LINE X(51)X(398)

Trilinears u tan A : v tan B : w tan C, where

u(A,B,C) = cos(B - C) + 2 cos(A - π/3)

X(463) lies on these lines: 2,3 51,398 184,397

X(465) = EULER LINE INTERCEPT OF LINE X(216)X(395)

Trilinears u cot A : v cot B : w cot C, where

u : 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) is the {X(2),X(3)}-harmonic conjugate of X(466). For a list of other harmonic conjugates of X(465), click More at the top of this page.

X(465) = complement of X(473)

------------------------------------------------------------------------

X(466) = EULER LINE INTERCEPT OF LINE X(216)X(396)

Trilinears u cot A : v cot B : w cot C, where

u : 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(466) is the {X(2),X(3)}-harmonic conjugate of X(465). For a list of other harmonic conjugates of X(466), click More at the top of this page.

X(446) = complement of X(472)

X(470) = X(15)-CROSS CONJUGATE OF X(298)

Trilinears sin(A + π/3) csc 2A : sin(B + π/3) csc 2B : sin(C + π/3) csc 2C

X(470) lies on these lines: 2,3 18,275 264,301 298,340 302,317 343,634 394,633

X(470) = inverse-in-orthocentroidal-circle of X(471)

X(470) = X(15)-cross conjugate of X(298)

X(470) = X(4)-Hirst inverse of X(471)

------------------------------------------------------------------------

X(471) = X(16)-CROSS CONJUGATE OF X(299)

Trilinears sin(A - π/3) csc 2A : sin(B - π/3) csc 2B : sin(C - π/3) csc 2C

X(471) lies on these lines: 2,3 17,275 264,300 299,340 303,317 343,633 394,634

X(471) = inverse-in-orthocentroidal-circle of X(470)

X(471) = X(16)-cross conjugate of X(299)

X(471) = X(4)-Hirst inverse of X(470)

------------------------------------------------------------------------

X(472) = X(62)-CROSS CONJUGATE OF X(303)

Trilinears cos(A + π/3) csc 2A : cos(B + π/3) csc 2B : cos(C + π/3) csc 2C

X(472) lies on these lines: 2,3 13,275 53,395 264,298 299,317 343,621 394,622

X(472) = inverse-in-orthocentroidal-circle of X(473)

X(472) = anticomplement of X(466)

X(472) = X(62)-cross conjugate of X(303)

------------------------------------------------------------------------

X(473) = X(61)-CROSS CONJUGATE OF X(302)

Trilinears cos(A - π/3) csc 2A : cos(B - π/3) csc 2B : cos(C - π/3) csc 2C

X(473) lies on these lines: 2,3 14,275 53,396 264,299 298,317 343,622 394,621

X(473) = inverse-in-orthocentroidal-circle of X(472)

X(473) = anticomplement of X(465)

X(473) = X(61)-cross conjugate of X(302)

X(530) = DIRECTION OF VECTOR AX + BX + CX, where X = X(13)

Trilinears f(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)

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(530) = isogonal conjugate of X(2378)

------------------------------------------------------------------------

X(531) = DIRECTION OF VECTOR AX + BX + CX, where X = X(14)

Trilinears f(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)

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(531) = isogonal conjugate of X(2379)

------------------------------------------------------------------------

X(532) = DIRECTION OF VECTOR AX + BX + CX, where X = X(17)

Trilinears f(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)

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(532) = isogonal conjugate of X(2380)

------------------------------------------------------------------------

X(533) = DIRECTION OF VECTOR AX + BX + CX, where X = X(18)

Trilinears f(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)

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(533) = isogonal conjugate of X(2381)

X(554) = INTERSECTION OF LINES X(1)X(30) AND X(14)X(226)

Trilinears sec(A/2) csc(A/2 + π/3) : sec(B/2) csc(B/2 + π/3) : sec(C/2) csc(C/2 + π/X(554) lies on these lines: 1,30 7,1082 14,226 75,299

X(559) = (X(1),X(15))-ANSWER TO QUESTION A

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)

X(559) lies on these lines: 1,3 14,226 299,319

X(616) = ANTICOMPLEMENT OF X(13)

Trilinears [SBSC - 2SA(a2 + sqr(3) area)]/a

X(616) lies on these lines: 2,13 3,299 4,627 14,148 15,532 20,633 30,298 69,74 302,381 303,549 489,2043 490,2044

X(616) = reflection of X(I) in X(J) for these (I,J): (13,618), (148,14), (617,99), (621,298)

X(616) = anticomplement of X(13)

X(616) = anticomplementary conjugate of X(621)

X(616) = X(298)-Ceva conjugate of X(2)

------------------------------------------------------------------------

X(617) = ANTICOMPLEMENT OF X(14)

Trilinears [SBSC - 2SA(a2 - sqr(3) area)]/a

X(617) lies on these lines: 2,14 3,298 4,628 13,148 16,533 20,634 30,299 69,74 302,549 303,381

X(617) = reflection of X(I) in X(J) for these (I,J): (14,619), (148,13), (616,99), (622,299)

X(617) = anticomplement of X(14)

X(617) = anticomplementary conjugate of X(622)

X(617) = X(299)-Ceva conjugate of X(2)

------------------------------------------------------------------------

X(618) = COMPLEMENT OF X(13)

Trilinears [2SBSC + 5SAa2 + 2 sqr(3)(b2 + c2) area]/a

X(618) lies on these lines: 2,13 3,635 5,629 14,99 15,298 30,623 39,395 61,627 140,630 141,542 396,532

X(618) = midpoint of X(I) and X(J) for these (I,J): (13,616), (14,99), (15,298)

X(618) = reflection of X(619) in X(620)

X(618) = complementary conjugate of X(623)

X(618) = X(2)-Ceva conjugate of X(396)

X(618) = crosspoint of X(2) and X(298)

------------------------------------------------------------------------

X(619) = COMPLEMENT OF X(14)

Trilinears [2SBSC + 5SAa2 - 2 sqr(3) (b2 + c2) area]/a

X(619) lies on these lines:

2,14 3,636 5,630 13,99 16,299 30,624 39,396 62,628 140,629 141,542 395,533

X(619) = midpoint of X(I) and X(J) for these (I,J): (13,99), (14,617), (16,299)

X(619) = reflection of X(618) in X(620)

X(619) = complementary conjugate of X(624)

X(619) = X(2)-Ceva conjugate of X(395)

X(619) = crosspoint of X(2) and X(299)

X(633) = ANTICOMPLEMENT OF X(61)

Trilinears (SBSC + 2 sqr(3) SA area)/a

X(633) lies on these lines: 2,18 3,298 4,69 5,299 14,636 17,623 20,616 140,302 141,398 343,471 394,470 397,524

X(633) = anticomplement of X(61)

X(633) = anticomplementary conjugate of X(627)

------------------------------------------------------------------------

X(634) = ANTICOMPLEMENT OF X(62)

Trilinears (SBSC - 2 sqr(3) SA area)/a

X(634) lies on these lines: 2,17 3,299 4,69 5,298 13,635 18,624 20,617 140,303 141,397 343,470 394,471 398,524

X(634) = reflection of X(I) in X(J) for these (I,J): (62,636), (61,635)

X(634) = anticomplement of X(62)

X(634) = anticomplementary conjugate of X(628)

------------------------------------------------------------------------

X(635) = COMPLEMENT OF X(61)

Trilinears [2SBSC + SAa2 + 2 sqr(3) (b2 + c2) area]/a

X(635) lies on these lines: 2,18 3,618 5,141 13,634 16,627 17,299 62,298 140,619 397,532

X(635) = midpoint of X(61) and X(633)

X(635) = complement of X(61)

X(635) = complementary conjugate of X(629)

X(636) = COMPLEMENT OF X(62)

Trilinears [2SBSC + SAa2 - 2 sqr(3) (b2 + c2) area]/a

X(636) lies on these lines: 2,17 3,619 5,141 14,633 15,628 18,298 61,299 140,618 398,533

X(636) = midpoint of X(62) and X(634)

X(636) = complement of X(62)

X(636) = complementary conjugate of X(630)

X(1080) = INTERCEPT OF EULER LINE AND LINE X(13)X(98)

Trilinears f(A,B,C) = csc(B - C) [sin 2B cos(C - É÷) sin(C - π/3) - sin 2C cos(B - É÷) sin(B - π/3)]

X(1080) lies on these lines: 2,3 13,98 14,262 183,622 298,511 325,621

------------------------------------------------------------------------

X(1081) = POINT ALPHECCA

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)

X(1081) lies on these lines: 1,30 7,559 13,226 75,298

X(1082) = POINT ALPHERATZ

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)

X(1082) lies on these lines: 1,3 7,554 13,226 298,319

X(1082) = isogonal conjugate of X(1251)

X(1094) = TRILINEAR SQUARE OF X(15)

Trilinears where f(A,B,C) = sin2(A + π/3)

X(1094) lies on these lines: 15,36 48,163

------------------------------------------------------------------------

X(1095) = TRILINEAR SQUARE OF X(16)

Trilinears f(A,B,C) = sin2(A - π/3)

X(1095) lies on these lines: 16,36 48,163

X(1276) = 2nd EVANS PERSPECTOR

Trilinears f(A,B,C) = sqrt(3/4) + sin(A + π/3) - sin(B + π/3) - sin(C + π/3)

= sqrt(3/4) + cos(A - π/6) - cos(B - π/6) - cos(C - π/6)

X(1276) lies on these lines: 1,15 4,9 14,484 63,616

X(1276) = inverse-in-Bevan-circle of X(1277) (noted by Peter J. C. Moses, Sept. 8, 2004)

X(1250) = ISOGONAL CONJUGATE OF X(1081)

Trilinears 1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1081)

X(1250) lies on these lines: 1,16 6,31 15,35 37,1251

X(1250) = isogonal conjugate of X(1081)

X(1250) = crosssum of X(1) and X(1277)

------------------------------------------------------------------------

X(1251) = ISOGONAL CONJUGATE OF X(1082)

Trilinears 1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1082)

X(1251) lies on these lines: 1,15 7,559 13,80 37,1250 55,199

X(1251) = isogonal conjugate of X(1082)

------------------------------------------------------------------------

X(1277) = 3rd EVANS PERSPECTOR

Trilinears f(A,B,C) = sqrt(3/4) - sin(A - π/3) + sin(B - π/3) + sin(C - π/3)

= sqrt(3/4) - cos(A + π/6) + cos(B + π/6) + cos(C + π/6)

X(1277) lies on these lines: 1,16 4,9 13,484 63,617

X(1277) = inverse-in-Bevan-circle of X(1276) (noted by Peter J. C. Moses, Sept. 8, 2004)

X(1493) = NAPOLEON CROSSSUM

Trilinears f(A,B,C) = csc(B + π/6)csc(C - π/6) + csc(C + π/6)csc(B - π/6)

= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (3 sin2A - cos2A)(3 sin B sin C - cos B cos C)

X(1493) lies on these lines:

3,54 5,539 49,143 110,1173 113,137 141,575 206,576

X(1493) = midpoint of X(54) and X(195)

X(1493) = isogonal conjugate of X(1487)

X(1493) = X(110)-Ceva conjugate of X(1510)

X(1493) = crosspoint of X(61) and X(62

X(1510) = NAPOLEON CROSSDIFFERENCE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc(B + π/6)csc(C - π/6) - csc(C + π/6)csc(B - π/6)

= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (3 sin2A - cos2A)(cos B sin C - sin B cos C)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1510) is the crossdifference of the Napoleon points, X(17) and X(18).

X(1510) lies on these lines: 30,511 110,1291

X(1510) = isogonal conjugate of X(930)

X(1510) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,137), (110,1493)

X(1510) = X(137)-cross conjugate of X(143)

X(1510) = crosspoint of X(I) and X(J) for these (I,J): (4,933), (110,1173)

X(1510) = crosssum of X(I) and X(J) for these (I,J): (140,523), (512,570)

X(1510) = crossdifference of any two points on line X(6)X(17)

------------------------------------------------------------------------

X(1511) = FERMAT CROSSSUM

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc(B + π/3)csc(C - π/3) + csc(C + π/3)csc(B - π/3)

= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin2A - 3 cos2A)(sin B sin C - 3 cos B cos C)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1511) lies on these lines:

2,265 3,74 24,1112 30,113 36,1464 125,128 141,542 146,376 184,974 186,323 214,960 249,842 389,1493

X(1511) = midpoint of X(I) and X(J) for these (I,J): (3,110), (74,399)

X(1511) = reflection of X(I) in X(J) for these (I,J): (125,140), (1539,113)

X(1511) = complementary conjugate of X(2072)

X(1511) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,1154), (110,526)

X(1511) = crosspoint of X(I) and X(J) for these (I,J): (2,340), (15,16)

X(1511) = crosssum of X(13) and X(14)

X(1652) = 4th EVANS PERSPECTOR

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sqrt(3/4) - sin(A + π/3) + sin(B + π/3) + sin(C + π/3)

= sqrt(3/4) - cos(A - π/6) + cos(B - π/6) + cos(C - π/6)

X(1652) is the perspector of the outer Napoleon triangle and the excentral triangle. (See also X(1276) and X(1277).)

X(1652) lies on these lines: 1,15 2,7 46,1277 396,554 559,1100

X(1652) = X(I)-Ceva conjugate of X(J) for these (I,J): (554,1), (2160,1653)

------------------------------------------------------------------------

X(1653) = 5th EVANS PERSPECTOR

Trilinears f(A,B,C) = sqrt(3/4) + sin(A - π/3) - sin(B - π/3) - sin(C - π/3)

= sqrt(3/4) + cos(A + π/6) - cos(B + π/6) - cos(C + π/6)

X(1653) is the perspector of the inner Napoleon triangle and the excentral triangle.

X(1653) lies on these lines: 1,16 2,7 46,1276 395,1081 1082,1100

X(1653) = X(I)-Ceva conjugate of X(J) for these (I,J): (1081,1), (2160,1652)

X(1653) = X(1081)-aleph conjugate of X(1653)

X(2962) = TRILINEAR PRODUCT X(17)*X(18)

Trilinears = csc(A + π/6) csc(A - π/6)

X(2962) lies on these lines:

19,2290 37,2963 82,1733 93,225 759,930 1087,1749 2216,2964

X(2962) = isogonal conjugate of X(2964)

------------------------------------------------------------------------

X(2963) = BARYCENTRIC PRODUCT X(17)*X(18)

Trilinears = sin A csc(A + π/6) csc(A - π/6)

X(2963) lies on these lines:

2,1225 5,2965 6,17 25,2934 37,2962 50,252 53,1487 93,393 111,930 141,2987 216,1989 230,251 566,2165 588,590 589,615 1879,2937

X(2963) = isogonal conjugate of X(1994)

X(2963) = cevapoint of X(590) and X(615)

X(2963) = X(570)-cross conjugate of X(6)

X(2963) = crosssum of X(I) and X(J) for these I,J: 6,195 49,2965

X(2964) = TRILINEAR PRODUCT X(61)*X(62)

Trilinears sin(A + π/6) sin(A - π/6)

X(2964) lies on these lines:

1,21 35,2361 36,1399 109,1393 162,2166 163,2179 484,580 2148,2180 2216,2962

X(2964) = isogonal conjugate of X(2962)

X(2964) = X(2216)-Ceva conjugate of X(1)

------------------------------------------------------------------------

X(2965) = BARYCENTRIC PRODUCT X(61)*X(62)

Trilinears sin A sin(A + π/6) sin(A - π/6)

X(2965) lies on these lines:

3,6 5,2963 53,112 230,1627 524,1238

X(2965) = X(I)-Ceva conjugate of X(J) for these I,J: 1173,184 1994,49

X(2965) = crosssum of X(6) and X(2937)

X(98) = TARRY POINT

X(99) = STEINER POINT

X(148) = STEINER POINT OF ANTICOMPLEMENTARY TRIANGLE

X(61) = ISOGONAL CONJUGATE OF X(17)

X(11) = FEUERBACH POINT

X(202) = X(1)-CEVA CONJUGATE OF X(15)

X(203) = X(1)-CEVA CONJUGATE OF X(16)

X(76) = 3rd BROCARD POINT

X(299) = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER

X(80) REFLECTION OF INCENTER IN FEUERBACH POINT

X(1080) = INTERCEPT OF EULER LINE AND LINE X(13)X(98)

X(1082) = POINT ALPHERATZ

X(1251) = ISOGONAL CONJUGATE OF X(1082)

X(303) ISOTOMIC CONJUGATE OF 2nd NAPOLEON POINT

X(616) = ANTICOMPLEMENT OF X(13)

X(617) = ANTICOMPLEMENT OF X(14)

The list

202

203

299

300

301

302

303

383

395

396

397

398

463

465

466

470

471

472

473

530

531

532

533

554

559

616

617

618

619

633

634

635

636

1080

1081

1082

1094

1095

1250

1251

1276

1277

1493

1510

1511

1652

1653

2962

2963

2964

2965