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 f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^{ -5}(b^{ -3} + c^{ -3}) - a^{ -4}(b^{ -4} + c^{ -4})
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
As the isogonal conjugate of a point on the circumcircle, X(696) lies on the line at infinity. The first trilinear coordinate has the form
a^{m-1}(b^{n} + c^{n}) - a^{n-1}(b^{m} + c^{m}).
If m and n are distinct integers, this form fits the definition of even polynomial center as in Clark Kimberling, "Functional equations associated with triangle geometry," Aequationes Mathematicae 45 (1993) 127-152. This form, perhaps appearing initially here (July 7, 2001) defines a triangle center for arbitrary distinct real numbers m and n. Selected even infinity and circumcircle points begin at X(696); odd ones begin at X(768).
Certain points of this type occur prior to this section. They are as follows:
X(538) = even (- 2, 0) infinity point
X(536) = even (- 1, 0) infinity point
X(519) = even (0, 1) infinity point
X(106) = even (0, 1) circumcircle point
X(524) = even (0, 2) infinity point
X(111) = even (0, 2) circumcircle point
X(518) = even (1, 2) infinity point
X(105) = even (1, 2) circumcircle point
X(674) = even (2, 3) infinity point
X(675) = even (2, 3) circumcircle point
X(511) = even (2, 4) infinity point
X(98) = even (2, 4) circumcircle point
X(696) lies on these lines: 30,511 313,561
X(696) = isogonal conjugate of X(697)