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.
Information from Kimberling's Encyclopedia of Triangle Centers |
Trilinears a/A : b/B : c/C
Barycentrics a^{2}/A : b^{2}/B : c^{2}/CThis point is the limit as r approaches 1 of the perspector of the r-Hofstadter triangle and ABC. See X(360) for details.
x^{2} + y^{2} + z^{2} + 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 x^{2} + y^{2} + z^{2} - 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(b^{2}/B + c^{2}/C - a^{2}/A) : b(c^{2}/C + a^{2}/A - b^{2}/B) : c(a^{2}/A + b^{2}/B - c^{2}/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)