X(69) = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE

Trilinears

\((cos A)/a2 : (cos B)/b2 : (cos C)/c2\)

\(bc(b2 + c2 - a2) : ca(c2 + a2 - b2) : ab(a2 + b2 - c2)\)

\(sec2(A/2) - csc2(A/2) : :\)

Barycentrics

\(cot A : cot B : cot C\)

\(b2 + c2 - a2 : c2 + a2 - b2 : a2 + b2 - c2\)

\(cot B + cot C - cot ω : :\)

\(cot B + cot C - cot A - cot ω : :\)

\(SA : SB : SC\)

Notes

Let A’B’C’ be the anticomplementary triangle. Let A″ be the inverse-in-anticomplementary-circle of A, and define B″ and C″ cyclically. The lines A’A″, B’B″, C’C″ concur in X(69). (Randy Hutson, February 10, 2016)

Let A’B’C’ be the anticomplementary triangle. Let A″ be the orthogonal projection of A’ on line BC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(69). (Randy Hutson, February 10, 2016)

Let A’B’C’ be the half-altitude triangle. Let A″ be the trilinear pole of line B’C’, and define B″ and C″ cyclically. Let A* be the trilinear pole of line B″C″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(69). (Randy Hutson, February 10, 2016)

Let A2B2C2 be the 2nd Conway triangle. Let A’ be the cevapoint of B2 and C2, and define B’ and C’ cyclically. The lines AA’, BB’, CC’ concur in X(69). (Randy Hutson, December 10, 2016)

X(69) is the barycentric multiplier for the MacBeath circumconic. (The barycentric product of X(69) and the circumcircle is the MacBeath circumconic.) (Randy Hutson, August 19, 2019)

Let OA be the circle centered at the A-vertex of the 1st Ehrmann triangle and passing through A; define OB and OC cyclically. X(69) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(69) lies on the Lucas cubic and these lines: 2,6 3,332 4,76 7,8 9,344 10,969 20,64 22,159 54,95 63,71 72,304 73,77 74,99 110,206 125,895 144,190 150,668 189,309 192,742 194,695 200,269 219,1332 248,287 263,308 265,328 274,443 290,670 297,393 347,664 350,497 404,1014 478,651 485,639 486,640 520,879 1225,2888 1369,3410

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

X(69) = reflection of X(i) in X(j) for these (i,j): (2,599), (4,1352), (6,141), (20,1350), (193,6), (895,125), (1351,5), (1353,140)

X(69) = isogonal conjugate of X(25)

X(69) = isotomic conjugate of X(4)

X(69) = complement of X(193)

X(69) = anticomplement of X(6)

X(69) = anticomplementary conjugate of X(2)

X(69) = circumcircle-inverse of X(5866)

X(69) = cyclocevian conjugate of X(253)

X(69) = X(i)-Ceva conjugate of X(j) for these (i,j): (76,2), (304,345), (314,75), (332,326)

X(69) = cevapoint of X(i) and X(j) for these (i,j): (2,20), (3,394), (6,159), (8,329), (63,78), (72,306), (125,525)

X(69) = X(i)-cross conjugate of X(j) for these (i,j): (3,2), (63,348), (72,63), (78,345), (125,525), (306,304), (307,75), (343,76)

X(69) = crosspoint of X(i) and X(j) for these (i,j): (2,2996), (76,305), (314,332)

X(69) = X(2)-Hirst inverse of X(325)

X(69) = X(i)-beth conjugate of X(j) for these (i,j): (69,77), (99,347), (314,7), (332,69), (645,69), (668,69)

X(69) = barycentric product of PU(37)

X(69) = bicentric sum of PU(132)

X(69) = midpoint of PU(132)

X(69) = perspector of the orthic-of-medial triangle and the reference triangle

X(69) = perspector of ABC and the pedal triangle of X(20)

X(69) = perspector of ABC and (reflection in X(2) of the pedal triangle of X(2))

X(69) = intersection of extended sides P(11)U(45) and U(11)P(45) of the trapezoid PU(11)PU(45)

X(69) = perspector of ABC and 4th extouch triangle

X(69) = antipode of X(287) in hyperbola {A,B,C,:ref:X(2) <X(2)>,:ref:X(69) <X(69)>}}

X(69) = trilinear pole of line X(441)X(525)

X(69) = pole wrt polar circle of trilinear polar of X(393) (line X(460)X(512))

X(69) = X(48)-isoconjugate (polar conjugate) of X(393)

X(69) = X(6)-isoconjugate of X(19)

X(69) = X(92)-isoconjugate of X(32)

X(69) = antigonal image of X(895)

X(69) = crosssum of X(i) and X(j) for these (i,j): (6,3053), (32,1974)

X(69) = perspector of ABC and the 2nd pedal triangle of X(3)

X(69) = crosspoint of X(6) and X(159) wrt both the excentral and tangential triangles

X(69) = crosspoint of X(2) and X(20) wrt both the excentral and anticomplementary triangles

X(69) = homothetic center of anticomplementary triangle and 2nd antipedal triangle of X(4) (i.e., of 1st and 2nd antipedal triangles of X(4))

X(69) = perspector of the complement of the polar circle

X(69) = perspector of the inconic with center X(3)

X(69) = pole, wrt de Longchamps circle, of trilinear polar of X(95)

X(69) = perspector of the extraversion triangles of X(7) and X(8)

X(69) = {X(2),:ref:X(6) <X(6)>}-harmonic conjugate of X(3618)

X(69) = perspector of ABC and anticomplement of submedial triangle

X(69) = perspector of ABC and mid-triangle of orthic and dual of orthic triangles

X(69) = perspector of ABC and cross-triangle of ABC and 2nd Brocard triangle

X(69) = perspector of 2nd Conway triangle and cross-triangle of ABC and 2nd Conway triangle

X(69) = Lucas-isogonal conjugate of X(376)

X(69) = anticevian-isogonal conjugate of X(2)

X(69) = inverse-in-MacBeath-circumconic of X(22151)

X(69) = {X(7),:ref:X(8) <X(8)>}-harmonic conjugate of X(75)

X(69) = intersection of van Aubel lines of outer and inner Vecten triangles

X(69) = orthic-isogonal conjugate of X(19583)

X(69) = X(4)-Ceva conjugate of X(19583)