X(84) = ISOGONAL CONJUGATE OF X(40)

Trilinears

\(1/(cos B + cos C - cos A - 1) : :\)

\(a^2[a^2 - (b - c)^2]^2 - (b - c)^2[a^2 - (b + c)^2]^2 : :\)

Notes

Let A’,B’,C’ be the excenters. The perpendiculars from B’ to AB and from C’ to AC meet in a point A″. Points B″ and C″ are determined cyclically. The hexyl triangle, A″B″C″, is perspective to ABC, and X(84) is the perspector.

Let A’B’C’ be the extouch triangle. Let A″ be the orthocenter of AB’C’, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(84). (Randy Hutson, September 14, 2016)

Let A1B1C1 be the 1st Conway triangle. Let A’ be the crosspoint of B1 and C1, and define B’ and C’ cyclically. The lines AA’, BB’, CC’ concur in X(84). (Randy Hutson, December 2, 2017)

See Dasari Naga Vijay Krishna, On a Conic Through Twelve Notable Points, Int. J. Adv. Math. and Mech. 7(2) (2019) 1-15.

X(84) lies on the Darboux cubic, the circumellipse with center X(9), and these lines: 1,221 3,9 4,57 7,946 8,20 21,285 33,603 36,90 46,80 58,990 64,3353 171,989 256,988 294,580 309,314 581,941 944,1000 2130,3345 3346,3472 3347,3355

X(84) = reflection of X(i) in X(j) for these (i,j): (40,1158), (1490,3)

X(84) = isogonal conjugate of X(40)

X(84) = isotomic conjugate of X(322)

X(84) = X(i)-Ceva conjugate of X(j) for these (i,j): (189,282), (280,1)

X(84) = X(i)-cross conjugate of X(j) for these (i,j): (19,57), (56,1)

X(84) = X(280)-aleph conjugate of X(84)

X(84) = X(i)-beth conjugate of X(j) for these (i,j): (271,3), (280,280), (285,84)

X(84) = X(68)-of-the-hexyl-triangle

X(84) = trilinear pole of line X(650)X(1459)

X(84) = perspector of ABC and the reflection in X(9) of the antipedal triangle of X(9)

X(84) = Danneels point of X(110)

X(84) = trilinear product of vertices of hexyl triangle (i.e., the extraversions of X(40))

X(84) = hexyl-isotomic conjugate of X(12717)

X(84) = orthologic center of hexyl triangle to inverse(n(hexyl triangle)); the reciprocal orthologic center is X(65)

X(84) = perspector of ABC and cross-triangle of extouch and Hutson-extouch triangles

X(84) = Cundy-Parry Phi transform of X(9)

X(84) = Cundy-Parry Psi transform of X(57)

X(84) = intouch-to-excentral similarity image of X(4)