X₃₀₄

ISOTOMIC CONJUGATE OF X₁₉

X₃₀₄ is the isotomic conjugate of X₁₉.

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

X₃₀₄ (ISOTOMIC CONJUGATE OF X₁₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 75}, {69, 72}

• Related Centers: X₁₉ (CLAWSON POINT), X₁ (INCENTER), X₇₅ (ISOTOMIC CONJUGATE OF INCENTER), X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE), X₇₂ (ISOGONAL CONJUGATE OF X₂₈)

Trilinears: \((\cot A)\csc^{2} A : (\cot B)\csc^{2} B : (\cot C)\csc^{2} C\)

Barycentrics: \((\cos A)\csc^{2} A : (\cos B)\csc^{2} B : (\cos C)\csc^{2} C\)

Search Value (6, 9, 13): \(6.190909895273\)

▸ Alternate Coordinates1 form

• Trilinears: \(\cos A \csc(A - \omega ) : \cos B \csc(B - \omega ) : \cos C \csc(C - \omega )\)

▸ Incident Central Lines19 lines

{1, 75}, {8, 3263}, {63, 1102}, {69, 72}, {76, 85}, {92, 561}, {255, 293}, {279, 346}, {305, 306}, {309, 322}, {337, 1565}, {341, 668}, {345, 348}, {662, 2172}, {742, 2176}, {799, 2349}, {811, 1895}, {1921, 3061}, {1958, 1973}

▸ Functional Relations10 relations

• X₃₀₄ = isogonal conjugate of X₁₉₇₃

• X₃₀₄ = isotomic conjugate of X₁₉

• X₃₀₄ = complement of X₂₁₂₁₆

• X₃₀₄ = anticomplement of X₁₆₅₈₃

• X₃₀₄ = cevapoint of X_i and X_j for these (i,j): (63, 326), (69, 345), (312, 322)

• X₃₀₄ = X_i-cross conjugate of X_j for these (i,j): (63, 75), (306, 69)

• X₃₀₄ = X_i-isoconjugate of X_j for these (i,j): (6, 25), (48, 1096), (92, 560)

• X₃₀₄ = polar conjugate of X₁₀₉₆

• X₃₀₄ = trilinear product of vertices of Gemini triangle 35

• X₃₀₄ = trilinear product of vertices of Gemini triangle 36

Notes

Let A_aB_aC_a, A_bB_bC_b, A_cB_cC_c be the A-, B-, and C-anti-altimedial triangles, resp. X₃₀₄ is the trilinear product B_a*C_a*C_b*A_b*A_c*B_c. (Randy Hutson, November 2, 2017)