X₅₉₄

ISOGONAL CONJUGATE OF X₅₉₃

X₅₉₄ is the isogonal conjugate of X₅₉₃.

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

X₅₉₄ (ISOGONAL CONJUGATE OF X₅₉₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {319, 894}

• Related Centers: X₃₁₉ (ISOTOMIC CONJUGATE OF X₇₉), X₈₉₄ (ISOGONAL CONJUGATE OF X₈₉₃)

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

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

Search Value (6, 9, 13): \(3.568078024947\)

▸ Incident Central Lines15 lines

{6, 8}, {7, 599}, {9, 80}, {10, 37}, {45, 346}, {53, 318}, {75, 141}, {100, 1030}, {210, 430}, {220, 281}, {313, 321}, {319, 524}, {519, 1100}, {572, 952}, {762, 1089}

▸ Functional Relations11 relations

• X₅₉₄ = midpoint of X₃₁₉ and X₈₉₄

• X₅₉₄ = isogonal conjugate of X₅₉₃

• X₅₉₄ = isotomic conjugate of X₁₅₀₉

• X₅₉₄ = anticomplement of X₁₇₀₄₅

• X₅₉₄ = crosspoint of X₁₀ and X₃₂₁

• X₅₉₄ = crosssum of X_i and X_j for these (i,j): (6, 595), (58, 1333)

• X₅₉₄ = barycentric square of X₁₀

• X₅₉₄ = perspector of ABC and cross-triangle of ABC and Gemini triangle 19

• X₅₉₄ = barycentric product of vertices of Gemini triangle 26

• X₅₉₄ = {X₆,X₈}-harmonic conjugate of X₁₇₃₆₂

• X₅₉₄ = {X₃₆₆₁,X₃₆₆₂}-harmonic conjugate of X₁₇₂₂₈

Notes

Let Eu_aEu_bEu_c be the Feuerbach triangle. Let L_a be the tangent to the nine-point circle at Eu_a, and define L_b and L_c cyclically. L_a, L_b, L_c are also tangent to the Steiner inellipse. Let A″, B″, C″ be the respective touchpoints. The lines AA″, BB″, CC″ concur in X₅₉₄. (Randy Hutson, October 15, 2018)

For a construction of X₅₉₄, see Dasari Naga Vijay Krishna, “On A Simple Construction of Triangle Centers X₈, X₁₉₇, X(K) & X₅₉₄”, Scientific Inquiry and Review, Vol. 2, Issue 3, July 2018.