HJB --- GMA --- UFF


Click here to access the list of all triangle centers.

Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon Run Macro Tool, select the center name from the list and, then, click on the vertices A, B and C successively.

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Download all construction files and macros: (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           csc 2A : csc 2B : csc 2C
Barycentrics    sec A : sec B : sec C

Let LA be the line through X(4) parallel to the internal bisector of angle A, and let
A' = BC∩LA. Define B' and C' cyclically.

Alexei Myakishev, "The M-Configuration of a Triangle," Forum Geometricorum 3 (2003) 135-144,

proves that the lines AA', BB', CC' concur in X(92). He notes that another construction follows from Proposition 2 of the article: let A1 be the midpoint of the arc BC of the circumcircle that passes through A, and let A2 be the point, other than A, in which the A-altitude meets the circumcircle. Let A" = A1A2∩BC. Define B" and C" cyclically. Then the lines AA", BB", CC" concur in X(92).

X(92) lies on these lines:
1,29    2,273    4,8    7,189    19,27    25,242    31,162    38,240    40,412    47,91    55,243    57,653    85,331    100,917    226,342    239,607    255,1087    257,297    264,306    304,561    406,1068    608,894

X(92) = isogonal conjugate of X(48)
X(92) = isotomic conjugate of X(63)
X(92) = anticomplement of X(1214)
X(92) = X(I)-Ceva conjugate of X(J) for these (I,J): (85,342), (264,318), (286,4), (331,273)
X(92) = cevapoint of X(I) and X(J) for these (I,J): (1,19), (4,281), (47,48), (196,278)
X(92) = X(I)-cross conjugate of X(J) for these (I,J): (1,75), (4,273), (19,158), (48,91), (226,2), (281,318)
X(92) = crosspoint of X(I) and X(J) for these (I,J): (85,309), (264,331)
X(92) = crossdifference of any two points on line X(810)X(822)
X(92) = X(275)-aleph conjugate of X(47)
X(92) = X(I)-beth conjugate of X(J) for these (I,J): (92,278), (312,329), (648,57)

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

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