X₄₂

CROSSPOINT OF INCENTER AND SYMMEDIAN POINT

X₄₂ is the crosspoint of incenter and symmedian point.

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

X₄₂ (CROSSPOINT OF INCENTER AND SYMMEDIAN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {321, 1215}

• Related Centers: X₃₂₁ (ISOTOMIC CONJUGATE OF X₈₁), X₁₂₁₅ (ISOGONAL CONJUGATE OF X₁₁₇₈)

Trilinears: \(a(b + c) : b(c + a) : c(a + b)\)

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

Tripolars: \((pending)\)

Search Value (6, 9, 13): \(1.283884145284\)

▸ Alternate Coordinates3 forms

• Trilinears: \((1 + \cos A)(\cos B + \cos C) : (1 + \cos B)(\cos C + \cos A) : (1 + \cos C)(\cos A + \cos B)\)

• Trilinears: \(a(ar - S) : b(br - S): c(cr - S)\)

• Trilinears: \(\csc B + \csc C : :\)

▸ Incident Central Lines29 lines

{1, 2}, {3, 967}, {6, 31}, {9, 941}, {25, 41}, {33, 393}, {35, 58}, {37, 210}, {38, 518}, {40, 581}, {48, 197}, {57, 1001}, {65, 73}, {81, 100}, {101, 111}, {165, 991}, {172, 199}, {181, 228}, {244, 354}, {308, 313}, {321, 740}, {517, 1064}, {560, 584}, {649, 788}, {694, 893}, {748, 1001}, {750, 940}, {894, 1045}, {942, 1066}

▸ Functional Relations25 relations

• X₄₂ = reflection of X₃₂₁ in X₁₂₁₅

• X₄₂ = isogonal conjugate of X₈₆

• X₄₂ = isotomic conjugate of X₃₁₀

• X₄₂ = X_i-Ceva conjugate of X_j for these (i,j): (1, 37), (6, 213), (10, 71), (55, 228)

• X₄₂ = crosspoint of X_i and X_j for these (i,j): (1, 6), (33, 55), (37, 65)

• X₄₂ = crosssum of X_i and X_j for these (i,j): (1, 2), (7, 77), (21, 81)

• X₄₂ = crossdifference of every pair of points on line X₂₃₉X₅₁₄

• X₄₂ = circumcircle-inverse of X₃₂₇₅₉

• X₄₂ = X₁-line conjugate of X₂₃₉

• X₄₂ = X_i-beth conjugate of X_j for these (i,j): (21, 551), (55, 42), (100, 226), (210, 210), (643, 171)

• X₄₂ = bicentric sum of PU(8)

• X₄₂ = PU(8)-harmonic conjugate of X₆₄₉

• X₄₂ = barycentric product of PU(32)

• X₄₂ = trilinear product of PU(85)

• X₄₂ = trilinear pole of line X₅₁₂X₇₉₈

• X₄₂ = Danneels point of X₁

• X₄₂ = {X₁,X₂}-harmonic conjugate of X₃₇₂₀

• X₄₂ = X₇₅-isoconjugate of X₅₈

• X₄₂ = X₉₂-isoconjugate of X₁₇₉₀

• X₄₂ = trilinear square root of X₈₇₂

• X₄₂ = perspector of ABC and unary cofactor triangle of 1st Conway triangle

• X₄₂ = perspector of ABC and unary cofactor triangle of 5th Conway triangle

• X₄₂ = perspector of unary cofactor triangles of 1st and 5th Conway triangles

• X₄₂ = perspector of ABC and unary cofactor triangle of Gemini triangle 2

• X₄₂ = barycentric product of vertices of Gemini triangle 15

Notes

Let A'B'C' be the extangents triangle. Let L_a be the trilinear polar of A', and define L_b and L_c cyclically. Let A″ = L_b∩L_c, B″ = L_c∩L_a, C″ = L_a∩L_b. The lines AA″, BB″, CC″ concur in X₄₂. (Randy Hutson, December 26, 2015)

Let A'B'C' be the extangents triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₄₂.(Randy Hutson, December 26, 2015)

Let A_b, A_c, B_c, B_a, C_a, C_b be as defined at X₃₅₈₈. Let A* be the intersection of the tangents to the Myakishev conic at B_a and C_a, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₄₂.(Randy Hutson, December 26, 2015)

X₄₂ is the {X₁,X₄₃}-harmonic conjugate of X₂.