## X(1157) (INVERSE-IN-CIRCUMCIRCLE OF X(54))

 Interactive Applet

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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           f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = a[(a2 - b2)2 - c2(a2 + b2)][(a2 - c2)2 - b2(c2 + a2)]U(a,b,c),
where U(a,b,c) = a6 - b6 - c6 + 3a2(b4 + c4 - a2b2 - a2c2) + b2c2(b2 + c2) - a2b2c2

Trilinears           g(A,B,C) : g(B,C,A) : g(C,A,B),
where g(A,B,C) = 4 cos A + cos 3A sec A sec(B - C)
[g(A,B,C) reported in Hyacinthos #7311, 6/23/03, N. Dergiades and D. Grinberg]

Barycentrics    af(a,b,c) : bf(b,c,a) : cf(c,a,b) = (sin A)g(A,B,C) : (sin B)g(B,C,A) : (sin C)g(C,A,B)

For any point X, let XA be the reflection of X in sideline BC, and define XB and XC cyclically. Then X(1157) is the unique point X for which the lines AXA, BXB, CXC concur on the circumcircle; the point of concurrence is X(1141).

X(1157) is the tangential of X(3) on the Neuberg cubic.

X(1157) lies on these lines: 3,54    5,252    30,1141    186,933

X(1157) = isogonal conjugate of X(1263)
X(1157) = inverse-in-circumcircle of X(54)

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