INSTITUTO DE MATEMÁTICA
HJB --- GMA --- UFF

X(165)
(CENTROID OF THE EXCENTRAL TRIANGLE)


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: 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           tan(B/2) + tan(C/2) - tan(A/2) : tan(C/2) + tan(A/2) - tan(B/2) : tan(A/2) + tan(B/2) - tan(C/2)
                                    = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a2 - 2a(b + c) - (b - c)2

Barycentrics    a[tan(B/2) + tan(C/2) - tan(A/2)] : b[tan(C/2) + tan(A/2) - tan(B/2)] : c[tan(A/2) + tan(B/2) - tan(C/2)]

X(165) = centroid of the triangle with vertices X(1), X(8), X(20)
X(165) = centroid of the triangle with vertices X(4), X(20), X(40)

X(165) lies on these lines:
1,3    2,516    4,1698    9,910    10,20    32,1571    42,991    43,573    63,100    71,610    105,1054    108,1767    109,212    164,167    166,168    191,1079    210,971    218,1190    220,1615    227,1394    255,1103    269,1253    355,550    371,1703    372,1702    376,515    380,579    386,1695    411,936    479,1323    498,1770    572,1051    574,1572    580,601    612,990    614,902    631,946    750,968    846,1719    950,1788    958,1706    962,11251011,1730    1342,1701    1343,1700

X(165) is the {X(3),X(40)}-harmonic conjugate of X(1). For a list of harmonic conjugates of X(165), click More at the top of this page.

X(165) = isogonal conjugate of X(3062)
X(165) = X(9)-Ceva conjugate of X(1)
X(165) = X(I)-aleph conjugate of X(J) for these (I,J):
(2,169), (9,165), (21,572), (100,101), (188,9), (259,43), (365,978), (366,57), (650,1053)

X(165) = X(I)-beth conjugate of X(J) for these (I,J): (100,165), (643,200)

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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