X(80) (REFLECTION OF INCENTER IN FEUERBACH POINT)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
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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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This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           1/(1 - 2 cos A) : 1/(1 - 2 cos B) : 1/(1 - 2 cos C)
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(b2 + c2 - a2 - bc)

Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/(b2 + c2 - a2 - bc)

X(80) lies on these lines:
1,5    2,214    7,150    8,149    9,528    10,21    30,484    33,1061    34,1063    36,104    40,90    46,84    65,79    313,314    497,1000    499,944    516,655    519,908    943,950

X(80) = midpoint of X(8) and X(149)
X(80) = reflection of X(I) in X(J) for these (I,J): (1,11), (100,10), (1317,1387)
X(80) = isogonal conjugate of X(36)
X(80) = isotomic conjugate of X(320)
X(80) = inverse-in-Fuhrmann-circle of X(1)
X(80) = anticomplement of X(214)
X(80) = cevapoint of X(10) and X(519)
X(80) = X(I)-cross conjugate of X(J) for these (I,J): (44,2), (517,1)
X(80) = X(8)-beth conjugate of X(100)

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