Interactive Applet |
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Information from Kimberling's Encyclopedia of Triangle Centers |
Trilinears a^{2}cos A : b^{2}cos B : c^{2}cos C
Barycentrics a^{3}cos A : b^{3}cos B : c^{3}cos CX(184) is the homothetic center of triangles ABC and A'B'C', the latter defined as follows: let B_{1} and C_{1} be the points where the perpendicular bisector BC meets sidelines CA and AB, and cyclically define C_{2}, A_{2}; A_{3}, B_{3}. Then A'B'C' is formed by the perpendicular bisectors of segments B_{1}C_{1}, C_{2}A_{2}, A_{3}B_{3}. (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_{c}C_{b}, XC_{a}A_{c}, XA_{b}B_{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 647,878X(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)