X₃₉

BROCARD MIDPOINT

X₃₉ 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. X₅₁₁, X₃₂, X₁₈₂.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₃₉ (BROCARD MIDPOINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {76, 194}

• Related Centers: X₇₆ (3rd BROCARD POINT), X₁₉₄ (-CEVA CONJUGATE OF X₂)

Trilinears: \(a(b^{2} + c^{2}) : b(c^{2} + a^{2}) : c(a^{2} + b^{2})\)

Barycentrics: \(a^{2}(b^{2} + c^{2}) : b^{2}(c^{2} + a^{2}) : c^{2}(a^{2} + b^{2})\)

Search Value (6, 9, 13): \(1.564479646463\)

▸ Alternate Coordinates6 forms

• Trilinears: \(\sin(A + \omega ) : \sin(B + \omega ) : \sin(C + \omega )\)

• Trilinears: \(\sin A + \sin(A + 2\omega ) : \sin B + \sin(B + 2\omega ) : \sin C + \sin(C + 2\omega )\)

• Trilinears: \(\cos A - \cos(A + 2\omega ) : \cos B - \cos(B + 2\omega ) : \cos C - \cos(C + 2\omega )\)

• Trilinears: \(\sin A + \cos A \tan \omega : :\)

• Trilinears: \(\cos A + \sin A \cot \omega : :\)

• Trilinears: \(a + 2R \cos A \tan \omega : :\)

▸ Linear Combinations2 combinations

• X₃₉ = \((r^{2} + 4rR - s^{2}) \cdot (r^{2} + 2rR + s^{2})\) X₁ - \(6rR(r^{2} + 4rR - s^{2})\) X₂ - \(2r^{2}(r^{2} + 4rR + s^{2})\) X₃ (Peter Moses, April 2, 2013)

• X₃₉ = \(P(1) + U(1)\)

X₃₉ lies on the bicevian conic of X₂ and X₉₉.

▸ Incident Central Lines25 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}

▸ Functional Relations58 relations

• X₃₉ = midpoint of X₇₆ and X₁₉₄

• X₃₉ = reflection of X₅₀₅₂ in X₆

• X₃₉ = isogonal conjugate of X₈₃

• X₃₉ = isotomic conjugate of X₃₀₈

• X₃₉ = complement of X₇₆

• X₃₉ = complementary conjugate of X₆₂₆

• X₃₉ = circumcircle-inverse of X₂₀₇₆

• X₃₉ = Brocard-circle-inverse of X₃₂

• X₃₉ = 1st-Lemoine-circle-inverse of X₂₄₅₈

• X₃₉ = antitomic conjugate of anticomplement of X₃₉₀₇₆

• X₃₉ = Steiner-circumellipse-inverse of anticomplement of X₃₉₇₈

• X₃₉ = eigencenter of anticevian triangle of X₅₁₂

• X₃₉ = X_i-Ceva conjugate of X_j for these (i,j): (2, 141), (3, 23208), (4, 211), (99, 512)

• X₃₉ = crosspoint of X_i and X_j for these (i,j): (2, 6), (141, 427)

• X₃₉ = crosssum of X_i and X_j for these (i,j): (2, 6), (251, 1176)

• X₃₉ = crossdifference of every pair of points on line X₂₃X₃₈₅

• X₃₉ = radical trace of 1st and 2nd Brocard circles

• X₃₉ = exsimilicenter of circles O(15,16) and O(61,62); the insimilicenter is X₃₂

• X₃₉ = radical trace of circles {X₁₅,X₆₂,PU(1)} and {X₁₆,X₆₁,PU(1)}

• X₃₉ = anticenter of cyclic quadrilateral PU(1)PU(39)

• X₃₉ = bicentric sum of PU(i) for these i: 1, 67

• X₃₉ = midpoint of PU(1)

• X₃₉ = PU(67)-harmonic conjugate of X₃₅₁

• X₃₉ = X₅₀₀₇ of 5th Brocard triangle

• X₃₉ = X₅₀₂₆ of 6th Brocard triangle

• X₃₉ = center of Moses circle

• X₃₉ = center of Gallatly circle

• X₃₉ = inverse-in-2nd-Brocard-circle of X₆

• X₃₉ = inverse-in-Kiepert-hyperbola of X₅

• X₃₉ = {X₆₁,X₆₂}-harmonic conjugate of X₅₇₅

• X₃₉ = {X₁₆₈₇,X₁₆₈₈}-harmonic conjugate of X₃₃₉₈

• X₃₉ = {X₂₀₀₉,X₂₀₁₀}-harmonic conjugate of X₅

• X₃₉ = Brocard axis intercept of radical axis of nine-point circles of ABC and circumsymmedial triangle

• X₃₉ = intersection of tangents to hyperbola {A, B, C,X₂,X₆}} at X₂ and X₆

• X₃₉ = perspector of circumconic centered at X₁₄₁

• X₃₉ = center of circumconic that is locus of trilinear poles of lines passing through X₁₄₁

• X₃₉ = trilinear pole, wrt medial triangle, of orthic axis

• X₃₉ = trilinear pole of line X₆₈₈X₃₀₀₅

• X₃₉ = perspector of medial triangle of ABC and medial triangle of 1st Brocard triangle

• X₃₉ = X₂₀₂₉-of-2nd-Brocard triangle

• X₃₉ = X₃₉-of-circumsymmedial-triangle

• X₃₉ = perspector, wrt symmedial triangle, of bicevian conic of X₆ and X₂₅

• X₃₉ = intersection of Brocard axes of ABC and 5th Euler triangle

• X₃₉ = X₉₂-isoconjugate of X₁₁₇₆

• X₃₉ = X₁₅₇₇-isoconjugate of X₈₂₇

• X₃₉ = eigencenter of Steiner triangle

• X₃₉ = perspector of ABC and unary cofactor triangle of circummedial triangle

• X₃₉ = center of (equilateral) unary cofactor triangle of Stammler triangle

• X₃₉ = X₇₇₅₃-of-4th-anti-Brocard-triangle

• X₃₉ = X₁₁-of-X₃PU(1)

• X₃₉ = X₁₁₅-of-X₃PU(1)

• X₃₉ = X₁₂₅-of-X₃PU(1)

• X₃₉ = homothetic center of Kosnita triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles

• X₃₉ = Cundy-Parry Phi transform of X₁₈₂

• X₃₉ = Cundy-Parry Psi transform of X₂₆₂

• X₃₉ = endo-similarity image of antipedal triangles of PU(1); the similitude center of these triangles is X₃

• X₃₉ = orthoptic-circle-of-Steiner-inellipse-inverse of X₃₂₅₂₆

• X₃₉ = QA-P42 (QA-Orthopole Center) of quadrangle ABCX₂ (see http://www.chrisvantienhoven.nl/quadrangle-objects/index.php/15-mathematics/encyclopedia-of-quadri-figures/quadrangle-objects/artikelen-qa/228-qa-p42.html)

Notes

The locus of the nine-point center in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is the circle through X₅ with center X₃₉. (Randy Hutson, August 29, 2018)

As illustrations of the Brocard porism mentioned just above, see

The Poncelet Homothetic Pair Contains an Aspect-Ratio Invariant Brocard Inellipse. Here the orbit of X₃₉ 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.

X₃₉ is the {X₃,X₆}-harmonic conjugate of X₃₂.