Trilinears           a²cos A : b²cos B : c²cos C
Barycentrics    a³cos A : b³cos B : c³cos C

X(184) is the homothetic center of triangles ABC and A'B'C', the latter defined as follows: let B₁ and C₁ be the points where the perpendicular bisector BC meets sidelines CA and AB, and cyclically define C₂, A₂; A₃, B₃. Then A'B'C' is formed by the perpendicular bisectors of segments B₁C₁, C₂A₂, A₃B₃. (Fred Lang, Hyacinthos #1190)

X(184) is the subject of Hyacinthos messages 5423-5441 (May, 2002). In #5423, Alexei Myakishev notes that X(184) serves as a common vertex of three triangles inside ABC, mutually congruent and similar to ABC. (The triangles can be labeled XB_cC_b, XC_aA_c, XA_bB_a, with B_c and C_b on side BC, C_a and A_c on side CA, and A_b and B_a on side AB.) See

Alexei Myakishev, On the Procircumcenter and Related Points , Forum Geometricorum 3 (2003) 29-34.

In #5435, Paul Yiu cites Fred Lang's construction of X(184) and notes that the three triangles are then easily constructed from X(184). The triangles determine three other triangles with common vertex X(184); in #5437, Nikos Dergiades notes that the vertex angles of these are 4A - π, 4B - π, 4C - π, and that

X(184) = X(63)-of-the-orthic-triangle = X(226)-of-the-tangential-triangle
X(184) = homothetic center of the orthic triangle and the medial triangle of the tangential triangle.

X(184) lies on these lines:
2,98    3,49    4,54    5,156    6,25    22,511    23,576    24,389    26,52    22,511    31,604    32,211    48,212    55,215    157,570    160,571    199,573    205,213    251,263    351,686    381,567    397,463    398,462    418,577    572,1011    647,878

X(184) is the {X(6),X(25)}-harmonic conjugate of X(51).

X(184) = isogonal conjugate of X(264)
X(184) = inverse-in-Brocard-circle of X(125)
X(184) = X(I)-Ceva conjugate of X(J) for these (I,J): (6,32), (54,6), (74,50)
X(184) = X(217)-cross conjugate of X(6)
X(184) = crosspoint of X(3) and X(6)

X(184) = crosssum of X(I) and X(J) for these (I,J): (2,4), (5,324), (6,157), (92,318), (273,342), (338,523), (339,850), (427,1235), (491,492)

X(184) = crossdifference of any two points on line X(297)X(525)
X(184) = X(32)-Hirst inverse of X(237)
X(184) = X(I)-beth conjugate of X(J) for these (I,J): (212,212), (692,184)