## X(14) (2ND ISOGONIC CENTER)

 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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This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           csc(A - π/3) : csc(B - π/3) : csc(C - π/3)
= sec(A + π/6) : sec(B + π/6) : sec(C + π/6)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a4 - 2(b2 - c2)2 + a2(b2 + c2 - 4*sqrt(3)*Area(ABC))

Construct the equilateral triangle BA'C having base BC and vertex A' on the positive side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA',BB',CC' concur in X(14). The antipedal triangle of X(14) is equilateral.

X(14) lies on these lines:
2,15    3,18    4,62    5,17    6,13    11,203    16,30    76,298    98,383    99,302    148,616    202,1478    226,554    262,1080    275,473    299,533    397,546    484,1276    530,671    532,622    633,636

X(14) is the {X(6),X(381)}-harmonic conjugate of X(13).

X(14) = reflection of X(I) in X(J) for these (I,J): (13,115), (16,395), (99,618), (299,624), (617,619)
X(14) = isogonal conjugate of X(16)
X(14) = isotomic conjugate of X(299)
X(14) = inverse-in-orthocentroidal-circle of X(13)
X(14) = complement of X(617)
X(14) = anticomplement of X(619)
X(14) = cevapoint of X(16) and X(61)
X(14) = X(I)-cross conjugate of X(J) for these (I,J): (16,17), (30,13), (395,2)

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