X₅₉₉

ISOTOMIC CONJUGATE OF X₅₉₈

X₅₉₉ is the isotomic conjugate of X₅₉₈.

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

X₅₉₉ (ISOTOMIC CONJUGATE OF X₅₉₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 69}

• Related Centers: X₂ (CENTROID), X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE)

Trilinears: \(bc(a^{2} - 2b^{2} - 2c^{2}) : ca(b^{2} - 2c^{2} - 2a^{2}) : ab(c^{2} - 2a^{2} - 2b^{2})\)

Barycentrics: \(a^{2} - 2b^{2} - 2c^{2} : b^{2} - 2c^{2} - 2a^{2} : c^{2} - 2a^{2} - 2b^{2}\)

Search Value (6, 9, 13): \(4.265829089912\)

▸ Incident Central Lines7 lines

{2, 6}, {3, 67}, {7, 594}, {8, 1086}, {76, 338}, {340, 458}, {381, 511}

▸ Functional Relations15 relations

• X₅₉₉ = midpoint of X₂ and X₆₉

• X₅₉₉ = reflection of X_i in X_j for these (i,j): (2, 141), (6, 2)

• X₅₉₉ = isogonal conjugate of X₁₃₈₃

• X₅₉₉ = isotomic conjugate of X₅₉₈

• X₅₉₉ = complement of X₁₉₉₂

• X₅₉₉ = anticomplement of X₅₉₇

• X₅₉₉ = crosssum of X₆ and X₁₃₈₄

• X₅₉₉ = X₆₇₁ of 1st Brocard triangle

• X₅₉₉ = Artzt-to-McCay similarity image of X₉₉

• X₅₉₉ = X₃-of-anti-Artzt-triangle

• X₅₉₉ = X₆-of-X₂-anti-altimedial triangle

• X₅₉₉ = 1st-Brocard-isogonal conjugate of X₅₀₂₆

• X₅₉₉ = 1st-Brocard-isotomic conjugate of X₅₄₃

• X₅₉₉ = center of pedal circle of X₂ wrt anticomplementary triangle

• X₅₉₉ = {X₂,X₁₉₉₂}-harmonic conjugate of X₅₉₇

Notes

Let O_A be the circle centered at the A-vertex of the Artzt triangle and passing through A; define O_B and O_C cyclically. X₅₉₉ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

Let O_A be the circle centered at the A-vertex of the antipedal triangle of X₆ and passing through A; define O_B and O_C cyclically. X₅₉₉ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)