X(58) (ISOGONAL CONJUGATE OF X(10))

 Interactive Applet

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Download all construction files and macros: tc.zip (10.1 Mb).
This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           a/(b + c) : b/(c + a) : c/(a + b)
Barycentrics    a2/(b + c) : b2/(c + a) : c2/(a + b)

X(58) is the point of concurrence of the Brocard axes of triangles BIC, CIA, AIB, ABC, (where I denotes the incenter, X(1)), as proved in

Antreas P. Hatzipolakis, Floor van Lamoen, Barry Wolk, and Paul Yiu, Concurrency of Four Euler Lines, Forum Geometricorum 1 (2001) 59-68.

X(58) lies on these lines:
1,21    2,540    3,6    7,272    8,996    9,975    10,171    20,387    25,967    27,270    28,34    29,162    35,42    36,60    40,601    41,609    43,979    46,998    56,222    65,109    82,596    84,990    86,238    87,978    99,727    101,172    103,112    106,110    229,244    269,1014    274,870    314,987    405,940    519,1043    942,1104    977,982    1019,1027

X(58) is the {X(3),X(6)}-harmonic conjugate of X(386).

X(58) = isogonal conjugate of X(10)
X(58) = isotomic conjugate of X(313)
X(58) = inverse-in-Brocard-circle of X(386)
X(58) = complement of X(1330)
X(58) = X(I)-Ceva conjugate of X(J) for these (I,J): (81,284), (267,501), (270,28)
X(58) = cevapoint of X(6) and X(31)
X(58) = X(I)-cross conjugate of X(J) for these (I,J): (6,81), (36,106), (56,28), (513,109)
X(58) = crosspoint of X(I) and X(J) for these (I,J): (1,267), (21,285), (27,86), (60,270)

X(58) = crosssum of X(I) and X(J) for these (I,J): (1,191), (6,199), (12,201), (37,210), (42,71), (65,227), (594,756)

X(58) = crossdifference of any two points on line X(523)X(661)
X(58) = X(6)-Hirst inverse of X(1326)
X(58) = X(I)-beth conjugate of X(J) for these (I,J): (21,21), (60,58), (110,58), (162,58), (643,58), (1098,283)

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

If you have questions or suggestions, please, contact us using the e-mail presented here.

Departamento de Matemática Aplicada -- Instituto de Matemática -- Universidade Federal Fluminense