X₃₀₉₈

INTERSECTION OF LINES X₃X_6 AND X₆₉X_74

Let AA₁A₂, BB₁B₂, CC₁C₂ be the circumcircle-inscribed equilateral triangles used in the construction of the Trinh triangle. Let A' be the midpoint of A₁ and A₂, and define B' and C' cyclically. Triangle A'B'C' is homothetic to ABC at X₃ and perspective to the Trinh triangle at X₃₀₉₈.

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

X₃₀₉₈ (INTERSECTION OF LINES X₃X_6 AND X₆₉X_74) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {69, 74}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE), X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT)

Trilinears: \(a(a^{4} - 2b^{4} - 2c^{4} - 2b^{2}c^{2} + c^{2}a^{2} + a^{2}b^{2}) : :\)

Search Value (6, 9, 13): \(9.677090093762\)

▸ Alternate Coordinates3 forms

• Trilinears: \(\sin A - 3 \cos A \cot \omega : :\)

• Trilinears: \(3 \cos A - \sin A \tan \omega : :\)

• Trilinears: \(2 \sin(A + 2\omega ) - \sin(A - 2\omega ) - \sin A : :\)

▸ Linear Combinations1 combination

• X₃₀₉₈ = \(3\) X₃ - X₆

▸ Incident Central Lines15 lines

{3, 6}, {4, 3096}, {20, 1352}, {22, 1495}, {30, 141}, {35, 1469}, {36, 3056}, {69, 74}, {184, 323}, {186, 1974}, {206, 1511}, {378, 1843}, {399, 2916}, {550, 1503}, {805, 842}

▸ Functional Relations10 relations

• X₃₀₉₈ = midpoint of X_i and X_j for these (i,j): (3, 1350), (20, 1352)

• X₃₀₉₈ = reflection of X_i in X_j for these (i,j): (182, 3), (576, 182), (1351, 575)

• X₃₀₉₈ = inverse-in-Brocard-circle of X₅₀₉₂

• X₃₀₉₈ = {X₃₇₁,X₃₇₂}-harmonic conjugate of X₅₀₀₇

• X₃₀₉₈ = inverse-in-circle-{{X₃₇₁,X₃₇₂,PU(1),PU(39)}} of X₅₀₀₇

• X₃₀₉₈ = X₇-of-Trinh-triangle if ABC is acute

• X₃₀₉₈ = Trinh-isotomic conjugate of X₃₃₅₇

• X₃₀₉₈ = X₁₈₂-of-circumcevian-triangle-of-X₅₁₁

• X₃₀₉₈ = X₁₈₂-of-5th-Brocard-triangle

• X₃₀₉₈ = 6th-Brocard-to-5th-Brocard similarity image of X₃