X₁₀₈₄

CROSSPOINT OF X₂ AND X₅₁₂

Let \(f(a,b,c) = a^{3}(b^{2} - c^{2})^{2}\). Then the line

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

X₁₀₈₄ (CROSSPOINT OF X₂ AND X₅₁₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {39, 597}, {2, 670}

• Related Centers: X₂ (CENTROID), X₅₁₂ (ISOGONAL CONJUGATE OF X₉₉), X₃₉ (BROCARD MIDPOINT), X₅₉₇ (MIDPOINT OF X₂ AND X₆), X₆₇₀ (TRILINEAR POLE OF LINE X₂X₃₉)

Trilinears: \(a^{3}(b^{2} - c^{2})^{2} : b^{3}(c^{2} - a^{2})^{2} : c^{3}(a^{2} - b^{2})^{2}\)

Barycentrics: \(a^{4}(b^{2} - c^{2})^{2} : b^{4}(c^{2} - a^{2})^{2} : c^{4}(a^{2} - b^{2})^{2}\)

Search Value (6, 9, 13): \(0.430418627157\)

X₁₀₈₄ lies on the Steiner inellipse.

▸ Incident Central Lines5 lines

{2, 670}, {6, 694}, {39, 597}, {115, 804}, {351, 865}

▸ Functional Relations22 relations

• X₁₀₈₄ = midpoint of X_i and X_j for these (i,j): (6, 694),(2, 3228)

• X₁₀₈₄ = reflection of X₃₅₀₇₃ in X₂

• X₁₀₈₄ = isogonal conjugate of X₃₄₅₃₇

• X₁₀₈₄ = complement of X₆₇₀

• X₁₀₈₄ = anticomplement of X₃₆₉₅₀

• X₁₀₈₄ = crosspoint of X_i and X_j for these {i,j}: {2, 512}, {6, 18105}, {32, 669}, {523, 6664}, {1974, 2489}, {798, 872}, {2395, 34238}

• X₁₀₈₄ = crosssum of X_i and X_j for these {i,j}: {2, 4576}, {6, 99}, {75, 21604}, {76, 670}, {110, 1627}, {305, 4563}, {799, 873}, {1509, 4623}, {2421, 5976}, {4631, 18021}, {5468, 31128}

• X₁₀₈₄ = crosssum of circumcircle intercepts of line X₂X₆

• X₁₀₈₄ = cevapoint of X₉₄₂₇ and X₂₃₂₁₆

• X₁₀₈₄ = trilinear pole of line X₁₆₄₅X₂₃₀₉₉

• X₁₀₈₄ = crossdifference of every pair of points on line X₉₉X₆₇₀

• X₁₀₈₄ = center of the circumconic {{A, B, C,X₂,X₆}}

• X₁₀₈₄ = projection from Steiner circumellipse to Steiner inellipse of X₃₂₂₈

• X₁₀₈₄ = Steiner-inellipse-antipode of X₃₅₀₇₃

• X₁₀₈₄ = perspector of circumconic centered at X₅₁₂ (parabola {{A, B, C,X₅₁₂,X₆₆₉}})

• X₁₀₈₄ = intersection of trilinear polars of X₅₁₂ and X₆₆₉

• X₁₀₈₄ = X₂-Ceva conjugate of X₅₁₂

• X₁₀₈₄ = perspector of ABC and the medial triangle of the cevian triangle of X₅₁₂

• X₁₀₈₄ = perspector of unary cofactor triangles of 3rd, 5th and 6th Brocard triangles

• X₁₀₈₄ = barycentric square of X₅₁₂

• X₁₀₈₄ = polar conjugate of isogonal conjugate of X₂₃₂₁₆

• X₁₀₈₄ = barycentric product X₃₂*X₁₁₅

Notes

f(a,b,c)x + f(b,c,a)y + f(c,a,b)z = 0 is tangent to the circumcircle at X₉₉.

Randy Hutson observed (9/23/2011) that the centers of homothety of the Lucas(L:W) homothetic triangles and triangle ABC form a circumconic which passes through the points X₄₉₃, X₄₉₄, X₅₈₈, and X₅₈₉. Indeed the perspectors are given by barycentric coordinates

\(a^{2}\)/[\(a^{2}\) + (L/W)S] : \(b^{2}\)/[\(b^{2}\) + (L/W)S] : \(c^{2}\)/[\(c^{2}\) + (L/W)S],

and the conic is the isogonal conjugate of the line X₂X₆. Thus, X₂, X₆, and dozens of other named points lie on the conic; click Tables at the top of ETC, select CENTRAL LINES, and scroll to #15.

X₁₀₈₄ is the center of the hyperbola H = {{A, B, C,X₂,X₆}}, which is tangent to Brocard axis at X₆ and to line X₂X₃₉ at X₂. Also, H is the locus of the trilinear pole of a line parallel to Lemoine axis (i.e. lines that pass through X₅₁₂), and H is the isotomic conjugate of line the X₂X₃₉. (Randy Hutson, July 20, 2016)