HJB --- GMA --- UFF


Click here to access the list of all triangle centers.

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 Run Macro Tool, select the center name from the list and, then, click on the vertices A, B and C successively.

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Download all construction files and macros: (10.1 Mb).
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) = bc[a4 + 2a3(b + c) - 4a2bc - (b + c)(b - c)2(2a + b + c)]
                                    = g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin B sin C + cos B + cos C - cos A - 1
                                    = h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = cos B + cos C + cos B cos C - 1
                                    = j(A,B,C) : j(B,C,A) : j(C,A,B), where j(A,B,C) = 1 - 2 cos2(B/2) cos2(C/2)

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)

X(962) is shown in

Michael S. Longuet-Higgins, "On the principal centers of a triangle," Elemente der Mathematik 56 (2001) 122-129

to complete a simple pattern of collinearities.

X(962) lies on these lines:
1,7    2,40    4,8    30,944    55,411    65,497    145,515    149,151    165,1125    278,412    382,952    392,443    484,499    942,1058

X(962) is the radical center of the circles centered at A, B, C, with respective
radii |CA| + |AB|, |AB| + |BC|, |BC| + |CA|. See

Floor van Lamoen, Problem 10734, American Mathematical Monthly 107 (2000) 658-659;

X(962) = reflection of X(I) in X(J) for these (I,J): (8,4), (20,1), (40,946), (944,1482)
X(962) = isogonal conjugate of X(963)
X(962) = anticomplement of X(40)
X(962) = X(309)-Ceva conjugate of X(2)

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

If you have questions or suggestions, please, contact us using the e-mail presented here.

Departamento de Matemática Aplicada -- Instituto de Matemática -- Universidade Federal Fluminense

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