X₃₀₆₉

INTERSECTION OF X₂X_6 AND X₄X_372

X₃₀₆₉ = complement of X₁₂₇₁

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

X₃₀₆₉ (INTERSECTION OF X₂X_6 AND X₄X_372) 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, 1271}

• Related Centers: X₂ (CENTROID), X₁₂₇₁ (ISOTOMIC CONJUGATE OF X₁₁₃₂)

Trilinears: \(\sin A - \sin B \sin C : :\)

Barycentrics: \((\sin A)(\sin A - \sin B \sin C) : :\)

Search Value (6, 9, 13): \(-0.620505304165\)

▸ Alternate Coordinates4 forms

• Trilinears: \((S - a^{2})/a : :\)

• Trilinears: \(\cos A - \sin A + \cos B \cos C : :\)

• Barycentrics: \(a^{2} - S : :\)

• Barycentrics: \(S_{B} + S_{C} - S : :\)

▸ Incident Central Lines20 lines

{1, 1123}, {2, 6}, {3, 1588}, {4, 372}, {5, 1587}, {20, 1152}, {32, 637}, {171, 605}, {216, 1589}, {238, 606}, {393, 494}, {489, 3053}, {577, 1590}, {589, 2165}, {894, 1267}, {920, 1336}, {946, 1703}, {1600, 1609}, {3074, 3076}, {3075, 3077}

▸ Functional Relations9 relations

• X₃₀₆₉ = isogonal conjugate of X₄₉₄

• X₃₀₆₉ = complement of X₁₂₇₁

• X₃₀₆₉ = X_i-Ceva conjugate of X_j for these (i,j): (393, 3068), (1586, 4)

• X₃₀₆₉ = crosspoint of X₂ and X₁₁₃₂

• X₃₀₆₉ = crosssum of X₆ and X₁₁₅₂

• X₃₀₆₉ = {X₂,X₆}-harmonic conjugate of X₃₀₆₈

• X₃₀₆₉ = X₂-of-2nd-tri-squares-triangle

• X₃₀₆₉ = X₂-of-2nd-tri-squares-central-triangle

• X₃₀₆₉ = homothetic center of ABC and 4th tri-squares central triangle

Notes

Let A'B'C' be the inner Vecten triangle. Let A″ be the trilinear pole of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₀₆₉. (Randy Hutson, July 21, 2017)