X₂₂₅₄

X₂-ISOCONJUGATE OF X₉₁₉

X₂₂₅₄ = cevapoint of X₆₆₅ and X₉₂₆

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

X₂₂₅₄ (X₂-ISOCONJUGATE OF X₉₁₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {103, 105}, {100, 109}

• Related Centers: X₁₀₃ (ANTIPODE OF X₁₀₁), X₁₀₅ (Λ(INCENTER, SYMMEDIAN POINT)), X₁₀₀ (ANTICOMPLEMENT OF FEUERBACH POINT), X₁₀₉ (Ψ(INCENTER, CIRCUMCENTER))

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

Search Value (6, 9, 13): \(-9.497924005520\)

▸ Incident Central Lines17 lines

{11, 244}, {44, 513}, {88, 1156}, {100, 109}, {103, 105}, {291, 812}, {294, 1027}, {514, 1734}, {522, 693}, {663, 905}, {665, 1642}, {690, 2292}, {764, 891}, {830, 1019}, {926, 1362}, {1054, 1768}, {3030, 6085}

▸ Functional Relations11 relations

• X₂₂₅₄ = reflection of X_i and X_j for these (i,j): (661, 1491), (663, 905)

• X₂₂₅₄ = isogonal conjugate of X₃₆₀₈₆

• X₂₂₅₄ = anticomplement of X₃₇₁₆

• X₂₂₅₄ = X_i-Ceva conjugate of X_j for these (i,j): (2, 38980), (291, 244), (673, 2170), (813, 38), (1025, 672), (1026, 518)

• X₂₂₅₄ = cevapoint of X₆₆₅ and X₉₂₆

• X₂₂₅₄ = crosspoint of X_i and X_j for these (i,j): (100, 1280), (513, 876), (518, 1026), (664, 673)

• X₂₂₅₄ = crosssum of X_i and X_j for these (i,j): (1, 2254), (105, 1027), (513, 1279), (663, 672), (1024, 2195)

• X₂₂₅₄ = X₆₆₆-aleph conjugate of X₈₁₂

• X₂₂₅₄ = radical center of {incircle, nine-point circle, Apollonius circle}

• X₂₂₅₄ = midpoint of Feuerbach hyperbola intercepts of antiorthic axis

• X₂₂₅₄ = perspector of hyperbola {{A, B, C,X₁,X₅₁₄}}

Notes

The tangent to the nine-point circle at the point where it is tangent to the incircle (X₁₁, the Feuerbach point) is here named the Feuerbach tangent line. It is line X₁₁X₂₄₄. The inverse of the incircle in the excircles radical circle is a circle tangent to the Apollonius circle at X₃₀₃₀. The line tangent to the Apollonius circle at X₃₀₃₀ is here named the Apollonius tangent line. It is line X₂₂₅₄X₃₀₃₀. X₂₂₅₄ is the intersection of the Feuerbach tangent line and the Apollonius tangent line. (Randy Hutson, January 15, 2019)