## X(768) (ODD (- 4, - 3) INFINITY POINT)

 Interactive Applet

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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(768) lies on the line at infinity. The first trilinear coordinate has the form

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

corresponding to an odd polynomial center in case m and n are distinct integers. See the note accompanying X(696), where even (m,n) infinity points and even (m,n) circumcircle points are introduced. [For nonzero n, "odd (m,n) circumcircle point" would be a misnomer (as the point is an even polynomial center); consequently, the prefix o- is used to distinguish this point from "even (m,n) circumcircle point" defined at X(696).] Certain points of these classes occur prior to this section. They are as follows:

X(523) = odd (- 4, - 2) infinity point
X(688) = odd (- 4, 0) infinity point
X(689) = o-(- 4, 0) circumcircle point
X(514) = odd (- 2, - 1) infinity point
X(101) = o-(- 2, - 1) circumcircle point
X(512) = odd (- 2, 0) infinity point
X(99) = o-(- 2, 0) circumcircle point
X(513) = odd (- 1, 0) infinity point
X(100) = o-(- 1, 0) circumcircle point
X(514) = odd (0, 1) infinity point
X(101) = o-(0, 1) circumcircle point
X(523) = odd (0, 2) infinity point
X(110) = o-(0, 2) circumcircle point
X(513) = odd (1, 2) infinity point
X(100) = o-(1, 2) circumcircle point
X(512) = odd (2, 4) infinity point
X(99) = o-(2, 4) circumcircle point

X(768) lies on this line: 30,511

X(768) = isogonal conjugate of X(769)

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