## X(31) (2ND POWER POINT)

 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 , 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           a2 : b2 : c2
= 1 - cos 2A : 1 - cos 2B : 1 - cos 2C

Barycentrics    a3 : b3 : c3

X(31) lies on these lines:
1,21    2,171    3,601    6,42    8,987    9,612    10,964    19,204    25,608    32,41    35,386    36,995    40,580    43,100    44,210    48,560    51,181    56,154    57,105    65,1104    72,976    75,82    76,734    91,1087    92,162    99,715    101,609    110,593    163,923    184,604    237,904    404,978    561,722    649,884    669,875    701,789    743,825    745,827    759,994    775,1097    937,1103    940,1001    999,1149    1139,1140

X(31) is the {X(1),X(63)}-harmonic conjugate of X(38).

X(31) = isogonal conjugate of X(75)
X(31) = isotomic conjugate of X(561)
X(31) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,48), (6,41), (9,205), (58,6), (82,1)
X(31) = X(213)-cross conjugate of X(6)
X(31) = crosspoint of X(I) and X(J) for these (I,J): (1,19), (6,56)

X(31) = crosssum of X(I) and X(J) for these (I,J): (1,63), (2,8), (7,347), (10,321), (239,1281), (244,514), (307,1441), (523,1086), (693,1111)

X(31) = crossdifference of any two points on line X(514)X(661)
X(31) = X(1403)-Hirst inverse of X(1428)
X(31) = X(I)-aleph conjugate of X(J) for these (I,J): (82,31), (83,75)
X(31) = X(I)-beth conjugate of X(J) for these (I,J): (21,993), (55,55), (109,31), (110,57), (643,31), (692,31)

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