X₄₇₄

EULER LINE INTERCEPT OF LINE X₁₀X_56

X₄₇₄ = complement of X₂₄₇₈

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

X₄₇₄ (EULER LINE INTERCEPT OF LINE X₁₀X_56) 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, 2478}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₂₄₇₈ (INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X), X₃ (CIRCUMCENTER)

Trilinears: \(\cos A - (a + b + c)/a : \cos B - (a + b + c)/b : \cos C - (a + b + c)/c\)

Barycentrics: \(a \cos A - (a + b + c) : b \cos B - (a + b + c) : c \cos C - (a + b + c)\)

Search Value (6, 9, 13): \(2.287909873541\)

▸ Alternate Coordinates1 form

• Trilinears: bcS - \(raS_{A}\) : caS - \(rbS_{B}\) : abS - \(rcS_{C})\)

▸ Incident Central Lines18 lines

{2, 3}, {8, 999}, {10, 56}, {35, 1001}, {36, 958}, {40, 392}, {46, 960}, {57, 72}, {65, 997}, {78, 942}, {142, 954}, {171, 978}, {183, 274}, {244, 976}, {283, 582}, {386, 940}, {579, 965}, {986, 1054}

▸ Functional Relations2 relations

• X₄₇₄ = complement of X₂₄₇₈

• X₄₇₄ = anticomplement of X₁₇₅₂₇

Notes

As a point on the Euler line, X₄₇₄ has Shinagawa coefficients \((abc\$a\$ - S^{2}, S^{2})\).