X₃₅₂₃

EULER LINE INTERCEPT OF LINE X₃₅X_390

X₃₅₂₃ = anticomplement of X₃₀₉₀

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

X₃₅₂₃ (EULER LINE INTERCEPT OF LINE X₃₅X_390) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {95, 253}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₉₅ (CEVAPOINT OF CENTROID AND CIRCUMCENTER), X₂₅₃ (-CROSS CONJUGATE OF X₂)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = 3 \cos A + \cos B \cos C\)

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(5.002508279179\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = \sec A + 3 \sec B \sec C\)

▸ Incident Central Lines22 lines

{1, 5265}, {2, 3}, {35, 390}, {36, 3085}, {40, 3306}, {84, 3305}, {95, 253}, {145, 1385}, {147, 620}, {153, 2551}, {165, 962}, {182, 193}, {389, 2979}, {391, 572}, {487, 1270}, {488, 1271}, {991, 3216}, {1151, 3069}, {1152, 3068}, {1155, 3485}, {1788, 2646}, {1189, 2888}

▸ Functional Relations6 relations

• X₃₅₂₃ = midpoint of X_i in X_j for these (i,j): (3, 3526), (3090, 3528)

• X₃₅₂₃ = reflection of X_i in X_j for these (i,j): (3090, 3526), (3528, 3)

• X₃₅₂₃ = anticomplement of X₃₀₉₀

• X₃₅₂₃ = circumcircle-inverse of X₃₇₉₄₅

• X₃₅₂₃ = homothetic center of dual of orthic triangle and (mid-triangle of orthic and circumorthic triangles)

• X₃₅₂₃ = Euler line intercept, other than X₂, of conic {{X₂,X₁₅,X₁₆,X₁₇,X₁₈}}

Notes

As a point on the Euler line, X₃₅₂₃ has Shinagawa coefficients \((3, -2)\).