X₁₇₂₄

MIMOSA TRANSFORM OF X₅₈

X₁₇₂₄ = crosspoint of X₁₆₂ and X₇₆₅

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

X₁₇₂₄ (MIMOSA TRANSFORM OF X₅₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 6}, {10, 31}

• Related Centers: X₁₆₂ (CEVAPOINT OF X₁₀₈ AND X₁₀₉), X₇₆₅ (CEVAPOINT OF X₁ AND X₁₀₀), X₁ (INCENTER), X₆ (SYMMEDIAN POINT), X₁₀ (SPIEKER CENTER), X₃₁ (2nd POWER POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = - (\cos A)/x + (\cos B)/y + (\cos C)/z\),\(x : y : z\) = X₅₈

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(-1.523028265713\)

▸ Alternate Coordinates1 form

• Trilinears: \(a^{2}(a + b + c) - bc(b + c) : b^{2}(a + b + c) - ca(c + a) : c^{2}(a + b + c) - ab(a + b)\) (M. Iliev, 5/13/07)

▸ Incident Central Lines34 lines

{1, 6}, {2, 58}, {4, 580}, {8, 595}, {10, 31}, {21, 286}, {28, 579}, {30, 582}, {32, 1009}, {34, 1708}, {35, 43}, {36, 978}, {46, 1707}, {47, 1737}, {83, 1008}, {90, 1039}, {109, 1788}, {171, 1698}, {191, 986}, {212, 950}, {226, 1451}, {255, 1210}, {270, 469}, {387, 452}, {515, 602}, {581, 1006}, {748, 1125}, {920, 1735}, {985, 1224}, {993, 1193}, {1020, 1398}, {1445, 1448}, {1726, 1829}, {1738, 1770}

▸ Functional Relations5 relations

• X₁₇₂₄ = X_i-Ceva conjugate of X_j for these (i,j): (28, 1), (579, 1754), (1778, 1707)

• X₁₇₂₄ = crosspoint of X₁₆₂ and X₇₆₅

• X₁₇₂₄ = crosssum of X₂₄₄ and X₆₅₆

• X₁₇₂₄ = X_i-aleph conjugate of X_j for these (i,j): (4, 1710), (27, 1730), (28, 1724), (266, 1047)

• X₁₇₂₄ = intercept, other than X₁, of line X₁X₆ and conic {{X₁,X₁₃,X₁₄,X₁₅,X₁₆}}