###### INSTITUTO DE MATEMÁTICA HJB --- GMA --- UFF

 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           a/A : b/B : c/C
Barycentrics    a2/A : b2/B : c2/C

This point is the limit as r approaches 1 of the perspector of the r-Hofstadter triangle and ABC. See X(360) for details.

x2 + y2 + z2 + yz(D + 1/D) + zx(E + 1/E) + xy(F + 1/F) = 0,

where D = cos A - sin A cot rA, E = cos B - sin B cot rB, F = cos C - sin C cot rC.

The Hofstadter ellipse E(1/2), given by x2 + y2 + z2 - 2yz - 2xz - 2xy = 0, passes through X(I) for these I: 244, 678, 2310, 2632, 2638, 2643.

Taking the limit as r tends to 0 gives information about the circumellipse, E(0) (which is also E(1)):

Equation:       ayz/A + bzx/B + cxy/C = 0
Center:       a(b2/B + c2/C - a2/A) : b(c2/C + a2/A - b2/B) : c(a2/A + b2/B - c2/C)
Intersection with circumcircle (other than A, B, C):       a/[A(B - C)] : b/[B(C - A)] : c/[C(A - B)].

X(359) = isogonal conjugate of X(360)

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