X₃₅₉₄

ISOGONAL CONJUGATE OF 2nd DIXIT POINT

X₃₅₉₄ is the isogonal conjugate of 2nd dixit point.

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

X₃₅₉₄ (ISOGONAL CONJUGATE OF 2nd DIXIT POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {486, 546}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₄₈₆ (INNER VECTEN POINT), X₅₄₆ (MIDPOINT OF X₄ AND X₅)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(b^{2} + c^{2} - a^{2} - 6 \operatorname{area}(ABC))\)

Barycentrics: \(a^{2}(2S_{A} - 3S)\) : \(b^{2}(2S_{B} - 3S)\) : \(c^{2}(2S_{C} - 3S)\) [Conway notation]

Search Value (6, 9, 13): \(-0.646170053813\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A, B, C) = 2 \cos A - 3 \sin A\)

▸ Incident Central Lines11 lines

{2, 3590}, {3, 6}, {486, 546}, {488, 3589}, {590, 3525}, {615, 1587}, {1588, 3529}, {3069, 3070}, {3071, 3146}, {3297, 3303}, {3298, 3304}

▸ Functional Relations2 relations

• X₃₅₉₄ = radical center of the Lucas(-3) circles

• X₃₅₉₄ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (3, 6, 3592), (6, 372, 1152), (6, 1152, 1151), (61, 62, 3312), (372, 3312, 6)