X(39) = BROCARD MIDPOINT

Trilinears

\(a(b2 + c2) : b(c2 + a2) : c(a2 + b2)\)

\(sin(A + ω) : sin(B + ω) : sin(C + ω)\)

\(sin A + sin(A + 2ω) : sin B + sin(B + 2ω) : sin C + sin(C + 2ω)\)

\(cos A - cos(A + 2ω) : cos B - cos(B + 2ω) : cos C - cos(C + 2ω)\)

\(sin A + cos A tan ω : :\)

\(cos A + sin A cot ω : :\)

\(a + 2R cos A tan ω : :\)

\(c/b : a/c : b/a and b/c : c/a : a/b. The third and fourth trilinear representations were given by Peter J. C. Moses (10/3/03); cf. :ref:`X(511) <X(511)>\), X(32), X(182). The locus of the nine-point center in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is the circle through X(5) with center X(39). (Randy Hutson, August 29, 2018) As illustrations of the Brocard porism mentioned just above, see`

Barycentrics

\(a2(b2 + c2) : b2(c2 + a2) : c2(a2 + b2)\)

Notes

X(39) is 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).

The locus of the nine-point center in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is the circle through X(5) with center X(39). (Randy Hutson, August 29, 2018)

As illustrations of the Brocard porism mentioned just above, see Brocard-Poncelet Porism, Stationary Brocard Points and Invariant Brocard Angle, Brocard Porism, Steiner Ellipses, and the Homothetic Poncelet Pair, (Dan Reznik and Ronaldo Garcia, August 9, 2020), The Poncelet Homothetic Pair Contains an Aspect-Ratio Invariant Brocard Inellipse. Here the orbit of X(39) is an ellipse. (Dan Reznik and Ronaldo Garcia, September 13, 2020)

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 10: The Brocard Points. X(39) lies on the bicevian conic of X(2) and X(99) and 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),:ref:X(6) <X(6)>}-harmonic conjugate of X(32). For a list of other harmonic conjugates of X(39), click Tables at the top of this page.

X(39) = midpoint of X(76) and X(194)

X(39) = reflection of X(5052) in X(6)

X(39) = isogonal conjugate of X(83)

X(39) = isotomic conjugate of X(308)

X(39) = complement of X(76)

X(39) = complementary conjugate of X(626)

X(39) = circumcircle-inverse of X(2076)

X(39) = Brocard-circle-inverse of X(32)

X(39) = 1st-Lemoine-circle-inverse of X(2458)

X(39) = antitomic conjugate of anticomplement of X(39076)

X(39) = Steiner-circumellipse-inverse of anticomplement of X(3978)

X(39) = eigencenter of anticevian triangle of X(512)

X(39) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,141), (3,23208), (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 every pair of points on line X(23)X(385)

X(39) = radical trace of 1st and 2nd Brocard circles

X(39) = exsimilicenter of circles O(15,16) and O(61,62); the insimilicenter is X(32)

X(39) = radical trace of circles {X(15),:ref:X(62) <X(62)>,PU(1)} and {X(16),:ref:X(61) <X(61)>,PU(1)}

X(39) = anticenter of cyclic quadrilateral PU(1)PU(39)

X(39) = bicentric sum of PU(i) for these i: 1, 67

X(39) = midpoint of PU(1)

X(39) = PU(67)-harmonic conjugate of X(351)

X(39) = X(5007) of 5th Brocard triangle

X(39) = X(5026) of 6th Brocard triangle

X(39) = center of Moses circle

X(39) = center of Gallatly circle

X(39) = inverse-in-2nd-Brocard-circle of X(6)

X(39) = inverse-in-Kiepert-hyperbola of X(5)

X(39) = {X(61),:ref:X(62) <X(62)>}-harmonic conjugate of X(575)

X(39) = {X(1687),:ref:X(1688) <X(1688)>}-harmonic conjugate of X(3398)

X(39) = {X(2009),:ref:X(2010) <X(2010)>}-harmonic conjugate of X(5)

X(39) = Brocard axis intercept of radical axis of nine-point circles of ABC and circumsymmedial triangle

X(39) = intersection of tangents to hyperbola {A,B,C,:ref:X(2) <X(2)>,:ref:X(6) <X(6)>}} at X(2) and X(6)

X(39) = perspector of circumconic centered at X(141)

X(39) = center of circumconic that is locus of trilinear poles of lines passing through X(141)

X(39) = trilinear pole, wrt medial triangle, of orthic axis

X(39) = trilinear pole of line X(688)X(3005)

X(39) = perspector of medial triangle of ABC and medial triangle of 1st Brocard triangle

X(39) = X(2029)-of-2nd-Brocard triangle

X(39) = X(39)-of-circumsymmedial-triangle

X(39) = perspector, wrt symmedial triangle, of bicevian conic of X(6) and X(25)

X(39) = intersection of Brocard axes of ABC and 5th Euler triangle

X(39) = X(92)-isoconjugate of X(1176)

X(39) = X(1577)-isoconjugate of X(827)

X(39) = eigencenter of Steiner triangle

X(39) = perspector of ABC and unary cofactor triangle of circummedial triangle

X(39) = center of (equilateral) unary cofactor triangle of Stammler triangle

X(39) = X(7753)-of-4th-anti-Brocard-triangle

X(39) = X(11)-of-:ref:`X(3) <X(3)>`PU(1)

X(39) = X(115)-of-:ref:`X(3) <X(3)>`PU(1)

X(39) = X(125)-of-:ref:`X(3) <X(3)>`PU(1)

X(39) = homothetic center of Kosnita triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles

X(39) = Cundy-Parry Phi transform of X(182)

X(39) = Cundy-Parry Psi transform of X(262)

X(39) = endo-similarity image of antipedal triangles of PU(1); the similitude center of these triangles is X(3)

X(39) = orthoptic-circle-of-Steiner-inellipse-inverse of X(32526)

X(39) = QA-P42 (QA-Orthopole Center) of quadrangle ABC:ref:X(2) <X(2)> (see http://www.chrisvantienhoven.nl/quadrangle-objects/index.php/15-mathematics/encyclopedia-of-quadri-figures/quadrangle-objects/artikelen-qa/228-qa-p42.html)