## X(696) (EVEN (- 4, - 3) INFINITY 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            f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b -3 + c -3) - a -4(b -4 + c -4)
Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(696) lies on the line at infinity. The first trilinear coordinate has the form

am-1(bn + cn) - an-1(bm + cm).

If m and n are distinct integers, this form fits the definition of even polynomial center as in Clark Kimberling, "Functional equations associated with triangle geometry," Aequationes Mathematicae 45 (1993) 127-152. This form, perhaps appearing initially here (July 7, 2001) defines a triangle center for arbitrary distinct real numbers m and n. Selected even infinity and circumcircle points begin at X(696); odd ones begin at X(768).

Certain points of this type occur prior to this section. They are as follows:

X(538) = even (- 2, 0) infinity point
X(536) = even (- 1, 0) infinity point
X(519) = even (0, 1) infinity point
X(106) = even (0, 1) circumcircle point
X(524) = even (0, 2) infinity point
X(111) = even (0, 2) circumcircle point
X(518) = even (1, 2) infinity point
X(105) = even (1, 2) circumcircle point
X(674) = even (2, 3) infinity point
X(675) = even (2, 3) circumcircle point
X(511) = even (2, 4) infinity point
X(98) = even (2, 4) circumcircle point

X(696) lies on these lines: 30,511    313,561

X(696) = isogonal conjugate of X(697)

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