## X(999) (MIDPOINT OF X(1) AND X(57))

 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            2 - cos A : 2 - cos B : 2 - cos C (Joe Goggins, 2002)
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + 4bc - b2 - c2)
Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(999) lies on these lines: 1,3    2,495    4,496    5,388    6,101    7,104    8,474    11,381    12,499    20,1058    30,497    63,392    77,1057    78,1059    81,859    145,404    329,405    376,390    518,997    527,551    601,1106    938,944    954,1006

X(999) is the {X(1),X(56)}-harmonic conjugate of X(3).

X(999) = midpoint of X(1) and X(57)
X(999) = isogonal conjugate of X(1000)
X(999) = X(89)-Ceva conjugate of X(6)
X(999) = crossdifference of any two points on line X(650)X(900)

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