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)