X₅₅₄

(X₁,X₁₃)-ANSWER TO QUESTION A

Let A' = (XY_A intersect BC), and define B' and C' cyclically. In

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

X₅₅₄ ((X₁,X₁₃)-ANSWER TO QUESTION A) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 30}, {14, 226}

• Related Centers: X₁ (INCENTER), X₃₀ (EULER INFINITY POINT), X₁₄ (2nd ISOGONIC CENTER), X₂₂₆ (-CEVA CONJUGATE OF X₆₅)

Trilinears: \(\sec(A/2) \csc(A/2 + \pi /3) : \sec(B/2) \csc(B/2 + \pi /3) : \sec(C/2) \csc(C/2 + \pi /3)\)

Barycentrics: \(\sin A \sec(A/2) \csc(A/2 + \pi /3) : \sin B \sec(B/2) \csc(B/2 + \pi /3) : \sin C \sec(C/2) \csc(C/2 + \pi /3)\)

Search Value (6, 9, 13): \(1.119721207656\)

▸ Alternate Coordinates1 form

• Barycentrics: \((a + b - c) \cdot (a - b + c) \cdot (a + 2 b + 2 c) + 2 \sqrt{3} a S : :\)

X₅₅₄ lies on the circumconic {A, B, C,X₂,X₇}}, the cubics K134 and K419b.

▸ Incident Central Lines22 lines

{1, 30}, {2, 33654}, {7, 1082}, {13, 43682}, {14, 226}, {55, 10652}, {57, 41225}, {75, 299}, {396, 1652}, {497, 30344}, {553, 37772}, {675, 36073}, {1086, 11073}, {1365, 18975}, {3475, 37830}, {3639, 3982}, {5239, 5249}, {5240, 5905}, {6186, 10648}, {7043, 11092}, {11705, 26700}, {30327, 37641}

▸ Functional Relations9 relations

• X₅₅₄ = isogonal conjugate of X₁₀₆₃₈

• X₅₅₄ = isotomic conjugate of X₄₀₇₁₄

• X₅₅₄ = X_i-cross conjugate of X_j for these (i,j): {553, 1081}, {30382, 85}

• X₅₅₄ = cevapoint of X_i and X_j for these (i,j): {1, 1652}, {33653, 33654}

• X₅₅₄ = X_i-Dao conjugate of X_j for these (i,j): {2, 40714}, {3, 10638}, {223, 559}, {7043, 40579}

• X₅₅₄ = X_i-isoconjugate of X_j for these (i,j): {1, 10638}, {16, 7126}, {31, 40714}, {35, 1251}, {55, 559}, {1250, 42677}, {2152, 7043}, {5239, 42624}, {5357, 19551}, {7005, 33653}, {7127, 7150}, {35057, 36072}

• X₅₅₄ = barycentric product X_i*X_j for these {i,j}: {75, 33654}, {85, 33653}, {301, 7051}, {1082, 30690}, {2307, 20565}, {3261, 36073}

• X₅₅₄ = barycentric quotient X_i/X_j for these {i,j}: {2, 40714}, {6, 10638}, {14, 7043}, {57, 559}, {1082, 3219}, {2154, 7126}, {2160, 1251}, {2306, 42677}, {2307, 35}, {5239, 44689}, {7051, 16}, {7052, 7150}, {11073, 19551}, {19373, 5357}, {33653, 9}, {33654, 1}, {33655, 46073}, {36073, 101}, {39152, 5240}, {40713, 42033}, {41225, 5239}

• X₅₅₄ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 4654, 1081}, {226, 3638, 559}, {3649, 37631, 1081}, {3782, 5434, 1081}

Notes

Suppose that X and Y are triangle centers. Let

Y_A = (Y of the triangle XBC),

• Y_B = (Y of the triangle XCA),

• Y_C = (Y of the triangle XAB).

Clark Kimberling, “Major Centers of Triangles,” Amer. Math. Monthly 104 (1997) 431-438,

Question A is this: for what choices of :g:`X <X>` and Y do the lines :g:`AA’ <[ A A’ ]>`, :g:`BB’ <[ B B’ ]>`, :g:`CC’ <[ C C’ ]>` concur? A solution (X,Y) will here be called the (X,Y)-answer to Question A. X₅₅₄ is the (X₁,X₁₃)-answer to Question A. (In the reference, see (9) on page 435, with Y = X₁₃.)