## X(35) ({X(1),X(3)}-HARMONIC CONJUGATE OF X(36))

 Interactive Applet

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 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           1 + 2 cos A : 1 + 2 cos B : 1 + 2 cos C
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2 + bc)

Barycentrics    sin A + sin 2A : sin B + sin 2B : sin C + sin 2C
= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 + c2 - a2 + bc)

Let A' be the inverse-in-circumcircle of the A-excenter, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(35).

X(35) lies on these lines:
1,3    4,498    8,993    9,90    10,21    11,140    12,30    22,612    24,33    31,386    34,378    37,267    42,58    43,1011    47,212    71,284    72,191    73,74    79,226    172,187    228,846    255,991    376,388    404,1125    411,516    474,1001    495,550    496,549    497,499    500,1154    595,902    950,1006    968,975    1124,1152

X(35) is the {X(1),X(3)}-harmonic conjugate of X(36).

X(35) = isogonal conjugate of X(79)
X(35) = inverse-in-circumcircle of X(484)
X(35) = X(500)-cross conjugate of X(1)
X(35) = crosssum of X(481) and X(482)
X(35) = X(943)-aleph conjugate of X(35)
X(35) = X(I)-beth conjugate of X(J) for these (I,J): (100,35), (643,10)

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