X₃₀₀₉

(LEMOINE AXIS)∩(NAGEL LINE)

X₃₀₀₉ = crossdifference of every pair of points on line X₂X₆₄₉

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

X₃₀₀₉ ((LEMOINE AXIS)∩(NAGEL LINE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 2}, {31, 172}

• Related Centers: X₃₂₂₆ (SS(a → bc) OF X₁₀₅), X₁ (INCENTER), X₂ (CENTROID), X₃₁ (2nd POWER POINT), X₁₇₂ (TRILINEAR PRODUCT X₆*X₁₇₁)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a[a(b^{2} + c^{2}) - bc(b + c)]\)

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

Search Value (6, 9, 13): \(-2.750146485765\)

▸ Incident Central Lines16 lines

{1, 2}, {31, 172}, {37, 1964}, {101, 2210}, {187, 237}, {192, 1740}, {213, 2308}, {238, 660}, {292, 672}, {536, 2234}, {694, 2054}, {741, 1931}, {872, 1100}, {1914, 2109}, {2111, 2116}, {2113, 2114}

▸ Functional Relations7 relations

• X₃₀₀₉ = isogonal conjugate of X₃₂₂₆

• X₃₀₀₉ = complement of X₂₀₃₅₂

• X₃₀₀₉ = anticomplement of X₂₀₃₄₀

• X₃₀₀₉ = X_i-Ceva conjugate of X_j for these (i,j): (238, 672), (660, 649), (727, 6), (1911, 42)

• X₃₀₀₉ = crosspoint of X_i and X_j for these (i,j): (1, 292), (6, 727), (1463, 1575)

• X₃₀₀₉ = crosssum of X_i and X_j for these (i,j): (1, 239), (2, 726)

• X₃₀₀₉ = crossdifference of every pair of points on line X₂X₆₄₉