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           1 + sec A/2 cos B/2 cos C/2 : 1 + sec B/2 cos C/2 cos A/2 : 1 + sec C/2 cos A/2 cos B/2
Barycentrics    g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)(1 + sec A/2 cos B/2 cos C/2)

If X is a point not between A and B, we make a detour of magnitude |AX| + |XB| - |AB| if we walk from A to B via X; then a point has the equal detour property if the magnitudes of the three detours, A to B via X, B to C via X, and C to A via X, are equal; X(176) is the only such point unless ABC has an angle greater than 2*arcsin(4/5), and then X(175) also has the equal detour property. Yff found that X(176) is also is the center of the inner Soddy circle. The following construction was found by Elkies: call two circles within ABC companion circles if they are the incircles of two triangles formed by dividing ABC into two smaller triangles by passing a line through one of the vertices and some point on the opposite side; chain of circles O(1), O(2), ... such that O(n),O(n+1) are companion incircles for every n consists of at most six distinct circles; there is a unique chain consisting of only three distinct circles; and for this chain, the three subdividing lines concur in X(176).

G. R. Veldkamp, "The isoperimetric point and the point(s) of equal detour," American Mathematical Monthly 92 (1985) 546-558.

Noam D. Elkies and Jiro Fukuta, Problem E 3236 and Solution, American Mathematical Monthly 97 (1990) 529-531 [proposed 1987].

X(176) lies on these lines: 1,7    8,1271    174,1143    226,1132    489,664    651,1124

X(176) = X(8)-Ceva conjugate of X(175)
X(176) = X(664)-beth conjugate of X(176)
X(176) = {X(1),X(7)}-harmonic conjugate of X(175)

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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