X₁₁₀₀

COMPLEMENT OF X₃₁₉

X₁₁₀₀ is the midpoint of the bicentric pair \(y : z : x\) and \(z : x : y\), where \(x : y : z\) = X₃₇

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

X₁₁₀₀ (COMPLEMENT OF X₃₁₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 319}

• Related Centers: X₂ (CENTROID), X₃₁₉ (ISOTOMIC CONJUGATE OF X₇₉)

Trilinears: \(2a + b + c : 2b + c + a : 2c + a + b\) (M. Iliev, 5/13/2007)

Barycentrics: \(a(2a + b + c) : b(2b + c + a) : c(2c + a + b)\)

Search Value (6, 9, 13): \(1.503900655143\)

▸ Alternate Coordinates1 form

• Trilinears: \(ar + S : br + S : cr + S\) (C. Lozada, 9/07/2013)

▸ Incident Central Lines14 lines

{1, 6}, {2, 319}, {36, 1030}, {48, 354}, {65, 604}, {71, 583}, {81, 593}, {86, 239}, {214, 1015}, {284, 501}, {517, 572}, {519, 594}, {536, 894}, {820, 836}

▸ Functional Relations16 relations

• X₁₁₀₀ = QA-P16 (QA-Harmonic Center) of quadrangle ABCX₁; see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/44-qa-p16.html

• X₁₁₀₀ = isogonal conjugate of X₁₂₅₅

• X₁₁₀₀ = isotomic conjugate of X₃₂₀₁₈

• X₁₁₀₀ = complement of X₃₁₉

• X₁₁₀₀ = anticomplement of X₁₇₂₃₉

• X₁₁₀₀ = crosspoint of X_i and X_j for these (i,j): (1, 81), (2, 79)

• X₁₁₀₀ = crosssum of X_i and X_j for these (i,j): (1, 37), (6, 35), (559, 1082)

• X₁₁₀₀ = crossdifference of every pair of points on line X₄₈₄X₅₁₃

• X₁₁₀₀ = bicentric sum of PU(31)

• X₁₁₀₀ = midpoint of PU(31)

• X₁₁₀₀ = reflection of X₃₇₇₅ in X₁₁₂₅

• X₁₁₀₀ = X₁-Ceva conjugate of X₁₉₆₂

• X₁₁₀₀ = X₁₉₆₂-Ceva conjugate of X₁₁₂₅ wrt incentral triangle

• X₁₁₀₀ = perspector wrt incentral triangle of bicevian conic of X₁ and X₂

• X₁₁₀₀ = polar conjugate of isogonal conjugate X₂₃₂₀₁

• X₁₁₀₀ = polar conjugate of isotomic conjugate of X₃₉₁₆

Notes

X₁₁₀₀ is the {X₁,X₆}-harmonic conjugate of X₃₇.