X₃₅₅₈

2nd PAPPUS POINT

X₃₅₅₈ = crosssum of PU(119)

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

X₃₅₅₈ (2nd PAPPUS POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {4, 3413}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₄ (ORTHOCENTER), X₃₄₁₃ (1st KIEPERT INFINITY POINT)

Trilinears: \(e \sin A - \sin(A - \omega ) : :\), where e is defined at X₁₃₄₀

Search Value (6, 9, 13): \(-7.928813637710\)

▸ Alternate Coordinates1 form

• Trilinears: \(a(a^{2} - Z)\), where \(Z^{2} = a^{4} + b^{4} + c^{4} - b^{2}c^{2} - c^{2}a^{2} - a^{2}b^{2}\)

▸ Linear Combinations2 combinations

• X₃₅₅₈ = \(2\) X₃ - \(3\) X₁₃₄₁

• X₃₅₅₈ = \(4\) X₃₉ - \(3\) X₁₃₄₀

X₃₅₅₈ lies on the cubics K028, K289, K704.

▸ Incident Central Lines4 lines

{3, 6}, {4, 3413}, {194, 3414}, {262, 6177}

▸ Functional Relations6 relations

• X₃₅₅₈ = reflection of X_i in X_j for these (i,j): (1380, 2028), (3557, 3095)

• X₃₅₅₈ = isogonal conjugate of X₆₁₇₈

• X₃₅₅₈ = crosssum of PU(119)

• X₃₅₅₈ = crosspoint of PU(119)

• X₃₅₅₈ = inverse-in-2nd-Brocard-circle of X₁₃₄₀

• X₃₅₅₈ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (3, 576, 3557), (6, 32, 3557), (6, 1380, 1340), (6, 2028, 1341), (61, 62, 3557), (371, 372, 1379), (1670, 1671, 1340), (1687, 1688, 2559), (1689, 1690, 3557), (3104, 3105, 3557)

Notes

X₃₅₅₈ is the crosspoint (and crosssum) of the imaginary foci of the Steiner inellipse. (Bernard Gibert, January 4, 2015)

X₃₅₅₈ is the Brocard axis intercept, other than X₁₃₄₁, of the circle {{X₁₃₄₁,PU(1)}}. Also, X₃₅₅₈ is the exsimilicenter of the 2nd Lemoine circle and the circle {{X₃₇₁,X₃₇₂,PU(1),PU(39)}}. (Randy Hutson, January 5, 2015)