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/(b + c) : b/(c + a) : c/(a + b)
Barycentrics a2/(b + c) : b2/(c + a) : c2/(a + b)X(58) is the point of concurrence of the Brocard axes of triangles BIC, CIA, AIB, ABC, (where I denotes the incenter, X(1)), as proved in
Antreas P. Hatzipolakis, Floor van Lamoen, Barry Wolk, and Paul Yiu, Concurrency of Four Euler Lines, Forum Geometricorum 1 (2001) 59-68.
X(58) lies on these lines:
1,21 2,540 3,6 7,272 8,996 9,975 10,171 20,387 25,967 27,270 28,34 29,162 35,42 36,60 40,601 41,609 43,979 46,998 56,222 65,109 82,596 84,990 86,238 87,978 99,727 101,172 103,112 106,110 229,244 269,1014 274,870 314,987 405,940 519,1043 942,1104 977,982 1019,1027X(58) is the {X(3),X(6)}-harmonic conjugate of X(386).
X(58) = isogonal conjugate of X(10)
X(58) = isotomic conjugate of X(313)
X(58) = inverse-in-Brocard-circle of X(386)
X(58) = complement of X(1330)
X(58) = X(I)-Ceva conjugate of X(J) for these (I,J): (81,284), (267,501), (270,28)
X(58) = cevapoint of X(6) and X(31)
X(58) = X(I)-cross conjugate of X(J) for these (I,J): (6,81), (36,106), (56,28), (513,109)
X(58) = crosspoint of X(I) and X(J) for these (I,J): (1,267), (21,285), (27,86), (60,270)X(58) = crosssum of X(I) and X(J) for these (I,J): (1,191), (6,199), (12,201), (37,210), (42,71), (65,227), (594,756)
X(58) = crossdifference of any two points on line X(523)X(661)
X(58) = X(6)-Hirst inverse of X(1326)
X(58) = X(I)-beth conjugate of X(J) for these (I,J): (21,21), (60,58), (110,58), (162,58), (643,58), (1098,283)