X₃₂₃₉

ISOGONAL CONJUGATE OF X₁₄₆₁

X₃₂₃₉ is the isogonal conjugate of X₁₄₆₁.

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

X₃₂₃₉ (ISOGONAL CONJUGATE OF X₁₄₆₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {9, 652}, {514, 661}

• Related Centers: X₁₄₆₁ (-BETH CONJUGATE OF X₆₅₁), X₉ (MITTENPUNKT), X₆₅₂ (CROSSDIFFERENCE OF X₁ AND X₄), X₅₁₄ (ISOGONAL CONJUGATE OF X₁₀₁), X₆₆₁ (CROSSDIFFERENCE OF X₁ AND X₂₁)

Trilinears: \((\cos B - \cos C)/(1 - \cos A) : :\)

Barycentrics: directed distance from A to Soddy line : :

Search Value (6, 9, 13): \(24.039943245611\)

▸ Alternate Coordinates1 form

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

X₃₂₃₉ lies on the Yff parabola.

▸ Incident Central Lines6 lines

{2, 2400}, {9, 652}, {100, 3234}, {101, 1309}, {111, 2760}, {514, 661}

▸ Functional Relations16 relations

• X₃₂₃₉ = isogonal conjugate of X₁₄₆₁

• X₃₂₃₉ = isotomic conjugate of X₆₅₈

• X₃₂₃₉ = complement of X₄₀₂₅

• X₃₂₃₉ = X_i-Ceva conjugate of X_j for these (i,j): (2, 2968), (190, 8), (341, 2310), (346, 1146), (644, 2321), (2399, 2804)

• X₃₂₃₉ = X_i-cross conjugate of X_j for these (i,j): (1146, 346), (2310, 341), (3119, 200)

• X₃₂₃₉ = crosspoint of X_i and X_j for these (i,j): (2, 1897), (8, 190), (644, 2321)

• X₃₂₃₉ = crosssum of X_i and X_j for these (i,j): (6, 1459), (56, 649), (513, 2262)

• X₃₂₃₉ = crossdifference of every pair of points on the line X₃₁X₅₆

• X₃₂₃₉ = trilinear pole of line X₁₁₄₆X₂₃₁₀

• X₃₂₃₉ = polar conjugate of X₃₆₁₁₈

• X₃₂₃₉ = pole wrt polar circle of trilinear polar of X₃₆₁₁₈ (line X₁₉X₅₇)

• X₃₂₃₉ = perspector of circumconic centered at X₂₉₆₈ (hyperbola {{A, B, C,X₈,X₇₅}})

• X₃₂₃₉ = center of circumconic that is locus of trilinear poles of lines passing through X₂₉₆₈

• X₃₂₃₉ = intersection of trilinear polars of X₈ and X₇₅

• X₃₂₃₉ = excentral-to-ABC barycentric image of X₆₅₀

• X₃₂₃₉ = perspector of Mandart hyperbola wrt extouch triangle