X₁₇₄₃

MIMOSA TRANSFORM OF X₂₂₂

X₁₇₄₃ = crosssum of X₂₄₄ and X₆₅₀

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

X₁₇₄₃ (MIMOSA TRANSFORM OF X₂₂₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 6}, {43, 165}

• Related Centers: X₁ (INCENTER), X₆ (SYMMEDIAN POINT), X₄₃ (-CEVA CONJUGATE OF X₁), X₁₆₅ (CENTROID OF THE EXCENTRAL TRIANGLE)

Trilinears: \(3a - b - c : :\) (M. Iliev, 5/13/2007)

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

Search Value (6, 9, 13): \(-0.525873758498\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\cot(B/2) + \cot(C/2) - \cot(A/2) : :\) (Randy Hutson, 9/23/2011)

• Trilinears: \(aS - rs^{2} : : (César Lozada, 9/07/2013)\)

▸ Incident Central Lines30 lines

{1, 6}, {10, 391}, {19, 1783}, {31, 200}, {36, 198}, {41, 572}, {43, 165}, {48, 1732}, {57, 1122}, {58, 936}, {71, 380}, {101, 604}, {169, 1046}, {173, 266}, {223, 1708}, {239, 1278}, {241, 1419}, {258, 259}, {269, 651}, {282, 1795}, {284, 1778}, {294, 1721}, {346, 519}, {579, 610}, {580, 1490}, {966, 1698}, {978, 1400}, {999, 1696}, {1249, 1785}, {1750, 1754}

▸ Functional Relations11 relations

• X₁₇₄₃ = isogonal conjugate of X₈₀₅₆

• X₁₇₄₃ = X_i-Ceva conjugate of X_j for these (i,j): (57, 1), (1476, 55)

• X₁₇₄₃ = crosspoint of X_i and X_j for these (i,j): (57, 1420), (651, 765)

• X₁₇₄₃ = crosssum of X₂₄₄ and X₆₅₀

• X₁₇₄₃ = {X₆,X₉}-harmonic conjugate of X₁

• X₁₇₄₃ = trilinear pole wrt excentral triangle of Gergonne line

• X₁₇₄₃ = perspector of ABC and unary cofactor triangle of Triangle T(-1,3)

• X₁₇₄₃ = excentral-isogonal conjugate of X₁₀₈₆₀

• X₁₇₄₃ = X₃₉₃-of-excentral-triangle

• X₁₇₄₃ = X_i-aleph conjugate of X_j for these (i,j): (1, 165), (2, 1766), (7, 169), (57, 1743), (81, 572), (174, 9), (259, 170), (266, 43), (365, 1742), (366, 40), (507, 164), (508, 63), (509, 1), (513, 1053), (651, 101)

• X₁₇₄₃ = perspector of excentral triangle and unary cofactor triangle of intangents triangle

Notes

Let A' be the center of the conic through the contact points of the B- and C-excircles with the sidelines of ABC. Define B' and C' cyclically. The triangle A'B'C' is perspective to the excentral triangle at X₁₇₄₃. See also X₆, X₂₅, X₂₁₈, X₂₂₂, X₉₄₀. (Randy Hutson, July 23, 2015)

In the plane of ABC, let a’ be the external bisector of A and a” its reflection in BC; define b” and c” cyclically. Then ABC and the triangle bounded by a”, b” and c” are perspective with perspector X₁₅₄₄₆. (César Lozada, October 10, 2018)