## X(648) (TRILINEAR POLE OF EULER LINE)

 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) = 1/[a(b2 - c2)(b2 + c2 - a2)]
= u(A,B,C) : u(B,C,A) : u(C,A,B), where u(A,B,C) = csc 2A csc(B - C)
Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(648) is constructed as the pole of the Euler line L as follows: let A", B", C" be the points where L meets the sidelines BC, CA, AB of the reference triangle ABC. Let A', B', C' be the harmonic conjugates of A", B", C" with respect to {B,C}, {C,A}, {A,B}, respectively, The lines AA', BB', CC' concur in X(648).

X(648) lies on these lines:
4,452    6,264    27,903    94,275    95,216    99,112    107,110    108,931    132,147    155,1093    162,190    185,1105    193,317    232,385    249,687    250,523    297,340    447,519    645,668    651,823    653,662    925,933    1020,1021    1075,1092

X(648) = reflection of X(I) in X(J) for these (I,J): (287,6), (340,297), (1494,2)
X(648) = isogonal conjugate of X(647)
X(648) = isotomic conjugate of X(525)
X(648) = cevapoint of X(110) and X(112)
X(648) = X(I)-cross conjugate of X(J) for these (I,J): (6,250), (110,99), (112,107), (520,95), (523,264)

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