X₁₁₅₄

ISOGONAL CONJUGATE OF X₁₁₄₁

X₁₁₅₄ is the isogonal conjugate of X₁₁₄₁.

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

X₁₁₅₄ (ISOGONAL CONJUGATE OF X₁₁₄₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {5, 51}, {3, 54}

• Related Centers: X₁₁₄₁ (GIBERT POINT), X₅ (NINE-POINT CENTER), X₅₁ (CENTROID OF ORTHIC TRIANGLE), X₃ (CIRCUMCENTER), X₅₄ (KOSNITA POINT)

Trilinears: \((1 - 4 \cos^{2} A)\cos(B - C) : (1 - 4 \cos^{2} B)\cos(C - A) : (1 - 4 \cos^{2} C)\cos(A - B)\)

Barycentrics: \(a(1 - 4 \cos^{2} A)\cos(B - C) : b(1 - 4 \cos^{2} B)\cos(C - A) : c(1 - 4 \cos^{2} C)\cos(A - B)\)

▸ Alternate Coordinates1 form

• Barycentrics: \(a^{2}[(a^{2} - b^{2} - c^{2})^{2} - b^{2}c^{2}](b^{4} + c^{4} - a^{2}b^{2} - a^{2}c^{2} - 2b^{2}c^{2}) : :\)

▸ Incident Central Lines11 lines

{2, 568}, {3, 54}, {4, 93}, {5, 51}, {26, 154}, {30, 511}, {35, 500}, {140, 389}, {185, 550}, {186, 323}, {403, 1112}

▸ Functional Relations9 relations

• X₁₁₅₄ = isogonal conjugate of X₁₁₄₁

• X₁₁₅₄ = complementary conjugate of X₁₂₈

• X₁₁₅₄ = X_i-Ceva conjugate of X_j for these (i,j): (3, 1511), (4, 128)

• X₁₁₅₄ = crosspoint of X_i and X_j for these (i,j): (5, 1263), (323, 340)

• X₁₁₅₄ = crosssum of X_i and X_j for these (i,j): (3, 539), (54, 1157)

• X₁₁₅₄ = X₃₀-of-orthic-triangle

• X₁₁₅₄ = X₃₀-of-tangential-triangle

• X₁₁₅₄ = excentral-to-ABC functional image of X₃₀

• X₁₁₅₄ = infinite point of tangent to hyperbola {{A, B, C,X₄,X₁₅}} at X₁₅ and tangent to hyperbola {{A, B, C,X₄,X₁₆}} at X₁₆

Notes

As the isogonal conjugate of a point on the circumcircle, X₁₁₅₄ lies on the line at infinity; X₁₁₅₄ is, in fact, the point where the Euler line of the orthic triangle meets the line at infinity (Bernard Gibert, Hyacinthos 1498, September 25, 2000).