## X(13) (1ST ISOGONIC CENTER (FERMAT POINT, TORRICELLI 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           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 negative 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(13). If each of the angles A, B, C is < 2*π/3, then X(13) is the only center X for which the angles BXC, CXA, AXB are equal, and X(13) minimizes the sum |AX| + |BX| + |CX|. The antipedal triangle of X(13) is equilateral.

The Evans conic is introduced in

Evans, Lawrence S., "A Conic Through Six Triangle Centers," Forum Geometricorum 2 (2002) 89-92.

X(13) lies on these lines:
2,16    3,17    4,61    5,18    6,14    11,202    15,30    76,299    80,1251    98,1080    99,303    148,617    203,1478    226,1081    262,383    275,472    298,532    484,1277    531,671    533,621    634,635

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

X(13) = reflection of X(I) in X(J) for these (I,J): (14,115), (15,396), (99,619), (298,623), (616,618)
X(13) = isogonal conjugate of X(15)
X(13) = isotomic conjugate of X(298)
X(13) = inverse-in-orthocentroidal-circle of X(14)
X(13) = complement of X(616)
X(13) = anticomplement of X(618)
X(13) = cevapoint of X(15) and X(62)
X(13) = X(I)-cross conjugate of X(J) for these (I,J): (15,18), (30,14), (396,2)

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