## X(184) (INVERSE OF X(125) IN THE BROCARD CIRCLE)

 Interactive Applet

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 Information from Kimberling's Encyclopedia of Triangle Centers

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 XBcCb, XCaAc, XAbBa, with Bc and Cb on side BC, Ca and Ac on side CA, and Ab and Ba 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)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.