X(1340) (INSIMILICENTER(CIRCUMCIRCLE, BROCARD CIRCLE))

 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.

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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 + 2(|OK| + R) cot ω cos A, where
|OK| = distance between X(3) and X(6),
ω = Brocard angle of ABC, and R = circumradius of ABC

Trilinears            g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = e cos A + cos(A - ω), e = (1 - 4 sin2ω)1/2

Barycentrics    af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1340) = internal center of similitude of circumcircle and Brocard circle (Peter J. C. Moses, 4/2003)

X(1340) lies on these lines: 2,1349    3,6    4,1348

X(1340) = inverse-in-Brocard-circle of X(1380)

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