X₃₇₆

CENTROID OF THE ANTIPEDAL TRIANGLE OF X₂

X₃₇₆ is the centroid of the antipedal triangle of X₂.

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

X₃₇₆ (CENTROID OF THE ANTIPEDAL TRIANGLE OF X₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 20} (Euler line)

• Related Centers: X₂ (CENTROID), X₂₀ (DE LONGCHAMPS POINT)

Trilinears: \(5 \cos A - \cos(B - C) : 5 \cos B - \cos(C - A) : 5 \cos C - \cos(A - B)\)

Barycentrics: \(5 \sin 2A - \sin 2B - \sin 2C : :\)

Search Value (6, 9, 13): \(10.935356995904\)

▸ Alternate Coordinates7 forms

• Trilinears: \(2 \cos A - \cos B \cos C : 2 \cos B - \cos C \cos A : 2 \cos C - \cos A \cos B\)

• Trilinears: \(3 \cos A - \sin B \sin C : 3 \cos B - \sin C \sin A : 3 \cos C - \sin A \sin B\)

• Trilinears: \(\sec A - 2 \sec B \sec C : \sec B - 2 \sec C \sec A : \sec C - 2 \sec A \sec B\)

• Trilinears: \((\csc A)(5 \sin 2A - \sin 2B - \sin 2C) : :\)

• Trilinears: \(a/(b \sec B + c \sec C) - 2 \cos A : :\)

• Barycentrics: \(2 S_{A} a^{2} - S_{B} S_{C} : :\)

• Barycentrics: \(5 a^{4} - 4 a^{2} (b^{2} + c^{2}) - (b^{2} - c^{2})^{2} : :\)

▸ Incident Central Lines22 lines

{1, 553}, {2, 3}, {35, 388}, {36, 497}, {40, 519}, {55, 1056}, {56, 1058}, {69, 74}, {98, 543}, {103, 544}, {104, 528}, {110, 541}, {112, 577}, {165, 515}, {316, 1007}, {390, 999}, {476, 841}, {477, 691}, {487, 490}, {488, 489}, {516, 551}, {524, 1350}

▸ Functional Relations24 relations

• X₃₇₆ = midpoint of X₂ and X₂₀

• X₃₇₆ = reflection of X_i in X_j for these (i,j): (2, 3), (4, 2), (381, 549)

• X₃₇₆ = isogonal conjugate of X₃₄₂₆

• X₃₇₆ = isotomic conjugate of X₃₆₈₈₉

• X₃₇₆ = circumcircle-inverse of X₇₄₆₄

• X₃₇₆ = orthocentroidal-circle-inverse of X₃₅₄₅

• X₃₇₆ = nine-point-circle-inverse of complement of X₃₇₉₅₂

• X₃₇₆ = complement of X₃₅₄₃

• X₃₇₆ = anticomplement of X₃₈₁

• X₃₇₆ = X₅₁-of-hexyl-triangle

• X₃₇₆ = centroid of circumcevian triangle of X₃₀

• X₃₇₆ = antipedal isogonal conjugate of X₂

• X₃₇₆ = {X₃,X₄}-harmonic conjugate of X₆₃₁

• X₃₇₆ = Thomson-isogonal conjugate of X₆

• X₃₇₆ = Lucas-isogonal conjugate of X₆₉

• X₃₇₆ = X₂-of-1st-anti-tri-squares-triangle

• X₃₇₆ = X₂-of-2nd-anti-tri-squares-triangle

• X₃₇₆ = homothetic center of 1st and 2nd anti-tri-squares triangles

• X₃₇₆ = insimilicenter of circumcircles of ABC and anti-Euler triangle; the exsimilicenter is X₄

• X₃₇₆ = {X₂₀₄₃,X₂₀₄₄}-harmonic conjugate of X₂₀

• X₃₇₆ = trisector nearest X₃ of segment X₃X₂₀

• X₃₇₆ = trisector nearest X₂₀ of segment X₄X₂₀

• X₃₇₆ = Ehrmann-mid-to-Johnson similarity image of X₂

• X₃₇₆ = Euler line intercept, other than X₄, of conic {X₄,X₁₃,X₁₄,X₁₅,X₁₆}}

Notes

As a point on the Euler line, X₃₇₆ has Shinagawa coefficients \((2, -3)\).

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, L_c cyclically. Let A' = L_b∩L_c, B' = L_c∩L_a, C' = L_a∩L_b. X₃₇₆ = X₆₉ of triangle A'B'C'. (Randy Hutson, July 20, 2016)

Let M_a be the polar of X₄ wrt the circle centered at A and passing through X₂, and define M_b, M_c cyclically. (Note: X₄ is the perspector of any circle centered at a vertex of ABC.) Let A″ = M_b∩M_c, and define B″, C″ cyclically. Triangle A″B″C″ is homothetic to ABC, and its orthocenter is X₃₇₆. (Randy Hutson, July 20, 2016)

Let A_a, A_b, A_c be the centers of the inverse-in-A-excircle of lines BC, CA, AB, resp. Let A' be the point of concurrence of lines AA_a, BA_b, CA_c. Define B', C' cyclically. Triangle A'B'C' is perspective to the excentral triangle at X₃₇₆. (Randy Hutson, July 20, 2016)

X₃₇₆ is the {X₃,X₂₀}-harmonic conjugate of X₄.