X₅₂₂

ISOGONAL CONJUGATE OF X₁₀₉

As the isogonal conjugate of a point on the circumcircle, X₅₂₂ lies on the line at infinity.

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

X₅₂₂ (ISOGONAL CONJUGATE OF X₁₀₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {30, 511}, {9, 657}

• Related Centers: X₁₀₉ (Ψ(INCENTER, CIRCUMCENTER)), X₃₀ (EULER INFINITY POINT), X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈), X₉ (MITTENPUNKT), X₆₅₇ (CROSSDIFFERENCE OF X₁ AND X₇)

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

Barycentrics: \(\cos B - \cos C : \cos C - \cos A : \cos A - \cos B\)

▸ Alternate Coordinates3 forms

• Trilinears: \(d(a,b,c) : :\), where d(a,b,c) = directed distance from A to line X₁X₃

• Barycentrics: \((b - c)(b + c - a) : :\)

• Barycentrics: \(\tan B/2 - \tan C/2 : :\)

▸ Incident Central Lines42 lines

{1, 1459}, {7, 2400}, {9, 657}, {11, 3326}, {30, 511}, {74, 2695}, {75, 3261}, {100, 655}, {101, 929}, {102, 2734}, {103, 2723}, {104, 2716}, {105, 2751}, {106, 2757}, {107, 2762}, {108, 2765}, {109, 1309}, {110, 2689}, {111, 2768}, {112, 2769}, {124, 2968}, {142, 3126}, {190, 666}, {240, 656}, {243, 652}, {649, 3509}, {650, 1639}, {663, 1944}, {664, 1275}, {693, 1266}, {1026, 2397}, {1027, 2402}, {1090, 2310}, {1292, 2730}, {1293, 2731}, {1294, 2732}, {1295, 2733}, {1296, 2735}, {1317, 3319}, {2490, 2496}, {2526, 3004}, {3063, 3287}

▸ Functional Relations25 relations

• X₅₂₂ = isogonal conjugate of X₁₀₉

• X₅₂₂ = isotomic conjugate of X₆₆₄

• X₅₂₂ = complementary conjugate of X₁₂₄

• X₅₂₂ = X_i-Ceva conjugate of X_j for these (i,j): (4, 124), (8, 11), (100, 10), (190, 9)

• X₅₂₂ = X₁₁-cross conjugate of X₈

• X₅₂₂ = crosspoint of X_i and X_j for these (i,j): (21, 100), (75, 190)

• X₅₂₂ = crosssum of X_i and X_j for these (i,j): (6, 663), (31, 649), (55, 652), (65, 513), (603, 1459), (692, 1415)

• X₅₂₂ = crossdifference of every pair of points on line X₆X₄₁

• X₅₂₂ = orthopoint of X₅₁₅

• X₅₂₂ = X_i-Hirst inverse of X_j for these (i,j): (514, 918), (519, 528)

• X₅₂₂ = isotomic conjugate of X₆₆₄

• X₅₂₂ = bicentric difference of PU(i) for these i: 20, 129

• X₅₂₂ = ideal point of PU(i) for these i: 20, 129

• X₅₂₂ = perspector of circumconic centered at X₁₁₄₆

• X₅₂₂ = center of circumconic that is locus of trilinear poles of lines passing through X₁₁₄₆

• X₅₂₂ = X₂-Ceva conjugate of X₁₁₄₆

• X₅₂₂ = trilinear pole of line X₁₁X₁₁₄₆

• X₅₂₂ = pole wrt polar circle of trilinear polar of X₆₅₃

• X₅₂₂ = X₄₈-isoconjugate (polar conjugate) of X₆₅₃

• X₅₂₂ = X₆-isoconjugate of X₆₅₁

• X₅₂₂ = extouch isogonal conjugate of X₁₃₆₄

• X₅₂₂ = Thomson-isogonal conjugate of X₁₀₂

• X₅₂₂ = Lucas-isogonal conjugate of X₁₀₂

• X₅₂₂ = barycentric square root of X₁₁₄₆

• X₅₂₂ = X₉₂-isoconjugate of X₃₂₆₆₀