## X(1113) (1ST EULER-LINE-CIRCUMCIRCLE INTERSECTION)

 Interactive Applet

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 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           (R - d)cos A - 2R cos B cos C : (R - d)cos B - 2R cos C cos A : (R - d)cos C - 2R cos A cos B,
where R = circumradius, d = distance |OH| between X(3) and X(4). (Joe Goggins, 2002)

Trilinears           f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = (1 - J) cos A - 2 cos B cos C, where J = |OH|/R; explicitly,
J = (1/abc)[S(6) - S(2,4) + 3a2b2c2]1/2, where S(6) = a6 + b6 + c6, and
S(2,4) = a2b4 + a2c4 + b2c4 + b2a4 + c2a4 + c2b4 (Peter J. C. Moses, 10/2/03)

Barycentrics    af(a,b,c) : bf(b,c,a) : cf(c,a,b)
= g(a,b,c) : g(b,c,a) : g(c,a,b),
where g(a,b,c) = 2RSBSC + (|OH| - R)a2SA,
|OH| = distance between X(3) and X(4), and R = circumradius (Peter J. C. Moses, 3/2003; cf. X(1313), X(1314))

X(1113) is a point of intersection of the Euler line and the circumcircle. The other is X(1114). Of the two, X(1113) is the one closer to X(4).

X(1113) = trilinear product X(110)*X(1823).

X(1113) lies on these lines: 2,3    109,1822

X(1113) = midpoint of X(4) and X(1312)
X(1113) = reflection of X(I) in X(J) for these (I,J): (4,1312), (1114,3)
X(1113) = anticomplement of X(1313)
X(1113) = X(250)-Ceva conjugate of X(1114)

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