X₃₅₃₄

EULER LINE INTERCEPT OF LINE X₁₅₄X_2777

X₃₅₃₄ = Stammler isogonal conjugate of X₆

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

X₃₅₃₄ (EULER LINE INTERCEPT OF LINE X₁₅₄X_2777) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {394, 399}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₃₉₄ (-CEVA CONJUGATE OF X₃), X₃₉₉ (PARRY REFLECTION POINT)

Trilinears: \(4 \sec A - 5 \sec B \sec C : :\)

Search Value (6, 9, 13): \(15.088351097612\)

▸ Alternate Coordinates1 form

• Trilinears: \(5 \cos A - 4 \cos B \cos C : :\)

▸ Linear Combinations1 combination

• X₃₅₃₄ = \(2\) X₂ - \(3\) X₃ = \(2\) X₃₈₁ - X₃₈₂

▸ Incident Central Lines10 lines

{2, 3}, {154, 2777}, {394, 399}, {542, 1350}, {590, 1327}, {599, 3098}, {615, 1328}, {999, 3058}, {1159, 3474}, {1384, 2549}

▸ Functional Relations6 relations

• X₃₅₃₄ = midpoint of X_i and X_j for these (i,j): (20, 376), (381, 1657), (3529, 3543)

• X₃₅₃₄ = reflection of X_i in X_j for these (i,j): (3, 376), (4, 549), (376, 550), (381, 3), (382, 381), (549, 548), (599, 3098), (3543, 5)

• X₃₅₃₄ = circumcircle-inverse of X₃₇₉₅₀

• X₃₅₃₄ = Stammler isogonal conjugate of X₆

• X₃₅₃₄ = {X₂,X₃}-harmonic conjugate of X₁₅₆₉₃

• X₃₅₃₄ = trisector nearest X₂₀ of segment X₃X₂₀

Notes

As a point on the Euler line, X₃₅₃₄ has Shinagawa coefficients \((5, -9)\).

Let M_aM_bM_c be the medial triangle. Let A″ be the pole of line M_bM_c wrt the A-power circle, and define B″, C″ cyclically. X₃₅₃₄ = X₂-of-A″B″C″. (Randy Hutson, March 14, 2018)

Let O_A be the circle centered at the A-vertex of the Gemini triangle 107 and passing through A; define O_B and O_C cyclically. X₃₅₃₄ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 28, 2020)