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 1/(cos A - 2 cos B cos C) : 1/(cos B - 2 cos C cos A) : 1/(cos C - 2 cos A cos B)
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[2a^{4} - (b^{2} - c^{2})^{2} - a^{2}(b^{2} + c^{2})]Barycentrics a/(cos A - 2 cos B cos C) : b/(cos B - 2 cos C cos A) : c/(cos C - 2 cos A cos B)
As the isogonal conjugate of the point in which the Euler line meets the line at infinity, X(74) lies on the circumcircle.
X(74) = circumcircle-antipode of X(110)
X(74) is the point of intersection, other than A, B, and C of the circumcircle and Jerabek hyperbola.In Hyacinthos 8129 (10/4/03), Floor van Lamoen noted that if X(74) is denoted by J, then each of the points A,B,C,J is J of the other three, in analogy with the well known property of orthocentric systems (that is, each of the points A,B,C,H is the orthocenter of the other three).
X(74) lies on these lines:
2,113 3,110 4,107 6,112 20,68 24,64 30,265 35,73 54,185 65,108 67,935 69,99 71,101 72,100 98,690 187,248 477,523 511,691 512,842 550,930X(74) = reflection of X(I) in X(J) for these (I,J): (4,125), (110,3), (146,113), (399,1511)
X(74) = isogonal conjugate of X(30)
X(74) = complement of X(146)
X(74) = anticomplement of X(113)
X(74) = cevapoint of X(I) and X(J) for these (I,J): (15,16), (50,184)
X(74) = crosssum of X(I) and X(J) for these (I,J): (3,399), (616),617)
X(74) = X(I)-cross conjugate of X(J) for these (I,J): (186,54), (526,110)