X₃₅₄₃

EULER LINE INTERCEPT OF LINE X₁₄₆X_148

X₃₅₄₃ = midpoint of X₂ and X₃₁₄₆

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

X₃₅₄₃ (EULER LINE INTERCEPT OF LINE X₁₄₆X_148) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3146}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₃₁₄₆ (-CEVA CONJUGATE OF X₂), X₃ (CIRCUMCENTER)

Trilinears: \(6 \cos B \cos C - \sin B \sin C : :\)

Search Value (6, 9, 13): \(-13.982607614342\)

▸ Linear Combinations1 combination

• X₃₅₄₃ = \(5\) X₂ - \(4\) X₃ = X₃₈₁ + \(3\) X₃₈₂

▸ Incident Central Lines19 lines

{2, 3}, {144, 3419}, {146, 148}, {147, 543}, {152, 544}, {153, 528}, {253, 317}, {371, 1131}, {372, 1132}, {388, 3058}, {390, 1478}, {393, 3163}, {515, 3241}, {519, 962}, {541, 3448}, {551, 1699}, {553, 938}, {671, 2794}, {1503, 1992}

▸ Functional Relations10 relations

• X₃₅₄₃ = midpoint of X₂ and X₃₁₄₆

• X₃₅₄₃ = reflection of X_i in X_j for these (i,j): (2, 4), (20, 2), (376, 381), (3529, 3534), (3534, 5)

• X₃₅₄₃ = anticomplement of X₃₇₆

• X₃₅₄₃ = circumcircle-inverse of X₃₇₉₅₂

• X₃₅₄₃ = harmonic center of polar and de Longchamps circles

• X₃₅₄₃ = intersection of tangents to conic {{X₄,X₁₃,X₁₄,X₁₅,X₁₆}} at X₁₃ and X₁₄

• X₃₅₄₃ = pole of Fermat axis wrt conic {{X₄,X₁₃,X₁₄,X₁₅,X₁₆}}

• X₃₅₄₃ = Euler line intercept, other than X₂₀, of conic {{X₁₃,X₁₄,X₁₅,X₁₆,X₂₀}}

• X₃₅₄₃ = {X₂,X₃}-harmonic conjugate of X₁₅₇₀₈

• X₃₅₄₃ = {X₃₈₁,X₃₈₂}-harmonic conjugate of X₁₅₆₈₄

Notes

As a point on the Euler line, X₃₅₄₃ has Shinagawa coefficients \((1, -6)\).

Let P_a be the parabola with focus A and directrix BC. Let L_a be the polar of X₃ wrt P_a. Define L_b and L_c cyclically. Let A' = L_b∩L_c, B' = L_c∩L_a, C' = L_a∩L_b. Then X₃₅₄₃ = X₁₉₃-of-A'B'C'. (Randy Hutson, September 5, 2015)