X₃₅₅₇

1st PAPPUS POINT

X₃₅₅₇ was submitted and called the Pappus point by Roland Bacher (March 11, 2009); coordinates and the related point X₃₅₅₈ were found by Peter Moses (March 12, 2009).

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

X₃₅₅₇ (1st 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, 3414}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₄ (ORTHOCENTER), X₃₄₁₄ (2nd KIEPERT INFINITY POINT)

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

Barycentrics: \(ag(a,b,c) : bg(b,c,a) : cg(c,a,b)\)

Search Value (6, 9, 13): \(0.622006435169\)

▸ Alternate Coordinates1 form

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

▸ Linear Combinations2 combinations

• X₃₅₅₇ = \(2\) X₃ - \(3\) X₁₃₄₀

• X₃₅₅₇ = \(4\) X₃₉ - \(3\) X₁₃₄₁

X₃₅₅₇ lies on the cubics K028, K289, K704.

▸ Incident Central Lines4 lines

{3, 6}, {4, 3414}, {194, 3413}, {262, 6178}

▸ Functional Relations6 relations

• X₃₅₅₇ = reflection of X_i in X_j for these (i,j): (1379, 2029), (3558, 3095)

• X₃₅₅₇ = isogonal conjugate of X₆₁₇₇

• X₃₅₅₇ = inverse-in-2nd-Brocard-circle of X₁₃₄₁

• X₃₅₅₇ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (3, 576, 3558), (6, 32, 3558), (6, 1379, 1341), (6, 2029, 1340), (61, 62, 3558), (371, 372, 1380), (1670, 1671, 1341), (1687, 1688, 2558), (1689, 1690, 3558), (3104, 3105, 3558)

• X₃₅₅₇ = crosssum of PU(118)

• X₃₅₅₇ = crosspoint of PU(118)

Notes

X₃₅₅₇ is the crosspoint (and crosssum) of the real foci of the Steiner inellipse. (Bernard Gibert, January 4, 2015)

X₃₅₅₇ is the Brocard axis intercept, other than X₁₃₄₀, of the circle {{X₁₃₄₀,PU(1)}}. Also, X₃₅₅₇ is the insimilicenter of the 2nd Lemoine circle and the circle {{X₃₇₁,X₃₇₂,PU(1),PU(39)}}. (Randy Hutson, January 5, 2015)