X₃₀₀₅

(DE LONGCHAMPS LINE)∩(LEMOINE AXIS)

Let L denote the line at infinity. Then (Lemoine axis) = (isogonal conjugate of isotomic conjugate of L) and (de Longchamps line) = (isotomic conjugate of isogonal conjugate of L), and X₃₀₀₅ = (Lemoine axis)∩(de Longchamps line). (Randy Hutson, February 10, 2016)

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

X₃₀₀₅ ((DE LONGCHAMPS LINE)∩(LEMOINE AXIS)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {187, 237}, {325, 523}

• Related Centers: X₁₈₇ (INVERSE-IN-CIRCUMCIRCLE OF X), X₂₃₇ (-LINE CONJUGATE OF X₂), X₃₂₅ (-HIRST INVERSE OF X₆₉), X₅₂₃ (ISOGONAL CONJUGATE OF X₁₁₀)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(b^{4} - c^{4})\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(-11.861814101453\)

X₃₀₀₅ lies on the bicevian conic of X₂ and X₅₁₂.

▸ Incident Central Lines7 lines

{2, 881}, {187, 237}, {325, 523}, {661, 756}, {689, 783}, {826, 2474}, {878, 2623}

▸ Functional Relations14 relations

• X₃₀₀₅ = reflection of X_i in X_j for these (i,j): (669, 647), (2528, 2525)

• X₃₀₀₅ = isogonal conjugate of X₄₅₇₇

• X₃₀₀₅ = isotomic conjugate of X₆₈₉

• X₃₀₀₅ = X_i-Ceva conjugate of X_j for these (i,j): (512, 688), (523, 826), (827, 6), (1634, 39)

• X₃₀₀₅ = X₂₅₃₁-cross conjugate of X₆₈₈

• X₃₀₀₅ = crosspoint of X_i and X_j for these (i,j): (6, 827), (39, 1634), (512, 523)

• X₃₀₀₅ = crosssum of X_i and X_j for these (i,j): (2, 826), (99, 110), (512, 1194)

• X₃₀₀₅ = complementary conjugate of X₇₆₆₈

• X₃₀₀₅ = complement of anticomplementary conjugate of X₃₉₃₄₆

• X₃₀₀₅ = crossdifference of every pair of points on line X₂X₃₂

• X₃₀₀₅ = perspector of circumconic centered at X₃₁₂₄

• X₃₀₀₅ = center of circumconic that is locus of trilinear poles of lines passing through X₃₁₂₄

• X₃₀₀₅ = X₂-Ceva conjugate of X₃₁₂₄

• X₃₀₀₅ = radical center of {circumcircle, Brocard circle, symmedial circle}