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[(a2 - b2)2 - c2(a2 + b2)][(a2 - c2)2 - b2(c2 + a2)]U(a,b,c),
where U(a,b,c) = a6 - b6 - c6 + 3a2(b4 + c4 - a2b2 - a2c2) + b2c2(b2 + c2) - a2b2c2
Trilinears g(A,B,C) : g(B,C,A) : g(C,A,B),
where g(A,B,C) = 4 cos A + cos 3A sec A sec(B - C)
[g(A,B,C) reported in Hyacinthos #7311, 6/23/03, N. Dergiades and D. Grinberg]
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b) = (sin A)g(A,B,C) : (sin B)g(B,C,A) : (sin C)g(C,A,B)
For any point X, let XA be the reflection of X in sideline BC, and define XB and XC cyclically. Then X(1157) is the unique point X for which the lines AXA, BXB, CXC concur on the circumcircle; the point of concurrence is X(1141).
X(1157) is the tangential of X(3) on the Neuberg cubic.
X(1157) lies on these lines: 3,54 5,252 30,1141 186,933
X(1157) = isogonal conjugate of X(1263)
X(1157) = inverse-in-circumcircle of X(54)