## X(1114) (2ND EULER-LINE-CIRCUMCIRCLE INTERSECTION)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon , select the center name from the list and, then, click on the vertices A, B and C successively.

 The JRE (Java Runtime Environment) is not enabled in your browser!

This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 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 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; see X(1113)

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)

X(1114) is a point of intersection of the Euler line and the circumcircle. Its antipode is X(1113).
X(1114) = trilinear product X(110)*X(1822).

X(1114) lies on these lines: 2,3    109,1823

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

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