## X(36) (INVERSE-IN-CIRCUMCIRCLE OF INCENTER)

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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)
= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec(A/2) cos(3A/2)

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)

X(36) is the {X(3),X(56)}-harmonic conjugate of X(1).

X(36) lies on these lines:
1,3    2,535    4,499    6,609    10,404    11,30    12,140    15,202    16,203    21,79    22,614    24,34    31,995    33,378    39,172    47,602    48,579    54,73    58,60    59,1110    63,997    80,104    84,90    99,350    100,519    101,672    106,901    109,953    187,1015    191,960    214,758    226,1006    238,513    255,1106    376,497    388,498    474,958    495,549    496,550    573,604    1030,1100

X(36) = midpoint of X(1) and X(484)
X(36) = reflection of X(I) in X(J) for these (I,J): (1,1319), (484,1155)
X(36) = isogonal conjugate of X(80)
X(36) = inverse-in-circumcircle of X(1)
X(36) = inverse-in-incircle of X(942)
X(36) = inverse-in-Bevan-circle of X(46)
X(36) = X(I)-Ceva conjugate of X(J) for these (I,J): (88,6), (104,1)
X(36) = crosspoint of X(58) and X(106)
X(36) = crosssum of X(I) and X(J) for these (I,J): (1,484), (10,519), (11,900)
X(36) = crossdifference of any two points on line X(37)X(650)
X(36) = X(104)-aleph conjugate of X(36)
X(36) = X(I)-beth conjugate of X(J) for these (I,J): (21,36), (100,36), (643,519)

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