## X(39) (BROCARD MIDPOINT)

 Interactive Applet

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 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.