HJB --- GMA --- UFF


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Interactive Applet

You can move the points A, B and C (click on the point and drag it).
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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 Run Macro Tool, select the center name from the list and, then, click on the vertices A, B and C successively.

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This applet was built with the free and multiplatform dynamic geometry software C.a.R..

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    615,641

X(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)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.

If you have questions or suggestions, please, contact us using the e-mail presented here.

Departamento de Matemática Aplicada -- Instituto de Matemática -- Universidade Federal Fluminense

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