## X(81) (CEVAPOINT OF INCENTER AND SYMMEDIAN POINT)

 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           1/(b + c) : 1/(c + a) : 1/(a + b)
Barycentrics    a/(b + c) : b/(c + a) : c/(a + b)

Let A'B'C' be the cevian triangle of X(1). Let A" be the symmedian point of triangle AB'C', and define B" and C" cyclically. Then the lines AA", BB", CC" concur in X(81). (Eric Danneels, Hyacinthos 7892, 9/13/03)

X(81) lies on these lines:
1,21    2,6    7,27    8,1010    19,969    28,60    29,189    32,980    42,100    43,750    55,1002    56,959    57,77    65,961    88,662    99,739    105,110    145,1043    226,651    239,274    314,321    377,387    386,404    411,581    593,757    715,932    859,957    941,967    982,985    1019,1022    1051,1054    1098,1104

X(81) = isogonal conjugate of X(37)
X(81) = isotomic conjugate of X(321)
X(81) = anticomplement of X(1211)
X(81) = X(I)-Ceva conjugate of X(J) for these (I,J): (7,229), (86,21), (286,28)
X(81) = cevapoint of X(I) and X(J) for these (I,J): (1,6), (57,222), (58,284)
X(81) = X(I)-cross conjugate of X(J) for these (I,J): (1,86), (3,272), (6,58), (57,27), (284,21)
X(81) = crosspoint of X(274) and X(286)
X(81) = crosssum of X(I) and X(J) for these (I,J): (1,846), (6,1030), (42,1334), (213,228)
X(81) = crossdifference of any two points on line X(512)X(661)
X(81) = X(I)-beth conjugate of X(J) for these (I,J): (333,333), (643,81), (645,81), (648,81), (662,81), (931,81)

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