X(64) = ISOGONAL CONJUGATE OF X(20)¶
Trilinears
\(1/(cos A - cos B cos C) : 1/(cos B - cos C cos A) : 1/(cos C - cos A cos B)\)
Barycentrics
\(a/(cos A - cos B cos C) : b/(cos B - cos C cos A) : c/(cos C - cos A cos B)\)
\(a^2/[3a^4 - 2a^2(b^2 + c^2) - (b^2 - c^2)^2] : : A construction of :ref:`X(64) <X(64)>\) appears in Lemoine’s 1886 paper cited at X(19). Let A’B’C’ be the half-altitude triangle. Let La be the trilinear polar of A’, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(64). (Randy Hutson, November 18, 2015) Let Oa be the circle with segment BC as diameter. Let A’ be the perspector of Oa. Let La be the polar of A’ wrt Oa. Define Lb and Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. The lines AA″, BB″, CC″ concur in X(64). (Randy Hutson, November 18, 2015) Let A’B’C’ be the cevian triangle of X(69). Let A″ be the orthocenter of AB’C’, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(64). (Randy Hutson, November 18, 2015) Let A’B’C’ be the reflection of the orthic triangle in X(4). Let A’’B’’C’’ be the trangential triangle, with respect ot the orthic triangle, of the circumconic of the orthic triangle with center X(4); i.e., the bicevian conic of X(4) and X(459). Then X(64) is the perspector of A’B’C’ and A’’B’’C’’. (Randy Hutson, November 18, 2015) The tangents at A, B, C to the Darboux cubic K004 concur in X(64). (Randy Hutson, November 18, 2015) X(64) lies on the Darboux cubic and these lines: 1,3182 3,154 4,3183 6,185 20,69 24,74 30,68 33,65 40,72 54,378 55,73 71,198 84,3353 265,382 3345,3472 3346,3355 X(64) = reflection of X(1498) in X(3) <X(3)>`
Notes
A construction of X(64) appears in Lemoine’s 1886 paper cited at X(19).
Let A’B’C’ be the half-altitude triangle. Let La be the trilinear polar of A’, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(64). (Randy Hutson, November 18, 2015)
Let Oa be the circle with segment BC as diameter. Let A’ be the perspector of Oa. Let La be the polar of A’ wrt Oa. Define Lb and Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. The lines AA″, BB″, CC″ concur in X(64). (Randy Hutson, November 18, 2015)
Let A’B’C’ be the cevian triangle of X(69). Let A″ be the orthocenter of AB’C’, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(64). (Randy Hutson, November 18, 2015)
Let A’B’C’ be the reflection of the orthic triangle in X(4). Let A’’B’’C’’ be the trangential triangle, with respect ot the orthic triangle, of the circumconic of the orthic triangle with center X(4); i.e., the bicevian conic of X(4) and X(459). Then X(64) is the perspector of A’B’C’ and A’’B’’C’’. (Randy Hutson, November 18, 2015)
The tangents at A, B, C to the Darboux cubic K004 concur in X(64). (Randy Hutson, November 18, 2015)
X(64) lies on the Darboux cubic and these lines: 1,3182 3,154 4,3183 6,185 20,69 24,74 30,68 33,65 40,72 54,378 55,73 71,198 84,3353 265,382 3345,3472 3346,3355
X(64) = reflection of X(1498) in X(3)
X(64) = isogonal conjugate of X(20)
X(64) = isotomic conjugate of X(14615)
X(64) = complement of X(6225)
X(64) = anticomplement of X(2883)
X(64) = circumcircle-inverse of X(11589)
X(64) = X(25)-cross conjugate of X(6)
X(64) = X(1)-beth conjugate of X(207)
X(64) = crosspoint of X(4) and X(3346)
X(64) = crosssum of X(3) and X(1498)
X(64) = perspector of hexyl triangle and anticevian triangle of X(2184)
X(64) = trilinear pole of line X(647)X(657)
X(64) = concurrence of normals to MacBeath circumconic at A, B, C
X(64) = isogonal conjugate, wrt tangential triangle of MacBeath circumconic (or anticevian triangle of X(3)), of X(1498)
X(64) = orthocenter of x(3):ref:X(6) <X(6)>X(2435)
X(64) = orthology center of ABC and half-altitude triangle
X(64) = intersection of tangents at X(3) and X(4) to Thomson cubic K002
X(64) = intersection of tangents at X(20) and X(64) to Darboux cubic K004
X(64) = perspector of ABC and the reflection in X(3) of the antipedal triangle of X(3) (tangential triangle)
X(64) = perspector of ABC and circumcircle antipode of circumanticevian triangle of X(3)
X(64) = perspector of ABC and unary cofactor triangle of half-altitude triangle
X(64) = X(2136)-of-orthic-triangle if ABC is acute
X(64) = X(8905)-of-excentral-triangle
X(64) = X(3)-vertex conjugate of X(3)