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(b2 + c2) : b(c2 + a2) : c(a2 + b2)
= sin(A + ω) : sin(B + ω) : sin(C + ω)
= sin A + sin(A + 2ω) : sin B + sin(B + 2ω) : sin C + sin(C + 2ω)
= cos A - cos(A + 2ω) : cos B - cos(B + 2ω) : cos C - cos(C + 2ω)Barycentrics a2(b2 + c2) : b2(c2 + a2) : c2(a2 + b2)
The midpoint of the 1st and 2nd Brocard points, given by trilinears c/b : a/c : b/a and b/c : c/a : a/b. The third and fourth trilinear representations were given by Peter J. C. Moses (10/3/03); cf. X(511), X(32), X(182).
X(39) lies on these lines:
1,291 2,76 3,6 4,232 5,114 9,978 10,730 36,172 37,596 51,237 54,248 83,99 110,755 140,230 141,732 185,217 213,672 325,626 395,618 493,494 512,881 588,589 590,642 597,1084 615,641X(39) is the {X(3),X(6)}-harmonic conjugate of X(32).
Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 10: The Brocard Points.
X(39) = midpoint of X(76) and X(194)
X(39) = isogonal conjugate of X(83)
X(39) = isotomic conjugate of X(308)
X(39) = inverse-in-Brocard-circle of X(32)
X(39) = inverse-in-1st-Lemoine-circle of X(2458)
X(39) = complement of X(76)
X(39) = complementary conjugate of X(626)
X(39) = eigencenter of anticevian triangle of X(512)
X(39) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,141), (4,211), (99,512)
X(39) = crosspoint of X(I) and X(J) for these (I,J): (2,6), (141,427)
X(39) = crosssum of X(I) and X(J) for these (I,J): (2,6), (251,1176)
X(39) = crossdifference of any two points on line X(661)X(830)