X₅₁

CENTROID OF ORTHIC TRIANGLE

X₅₁ is the centroid of orthic triangle.

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

X₅₁ (CENTROID OF ORTHIC TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Orthic Triangle △H_aH_bH_c — vertices H_a, H_b, H_c (Pedal triangle of Orthocenter X₄ (altitude feet))

• Concurrence Lines: H_a–X₅₁, H_b–X₅₁, H_c–X₅₁ concurring at X₅₁

Trilinears: \(a^{2}\cos(B - C) : :\)

Barycentrics: \(a^{3}\cos(B - C) : b^{3}\cos(C - A) : c^{3}\cos(A - B)\)

Search Value (6, 9, 13): \(0.120625582707\)

▸ Alternate Coordinates3 forms

• Trilinears: \(a[a^{2}(b^{2} + c^{2}) - (b^{2} - c^{2})^{2}] : :\)

• Trilinears: \(\sin A (\sin 2B + \sin 2C) : :\)

• Trilinears: \(\sec A (\csc 2B + \csc 2C) : :\)

▸ Linear Combinations1 combination

• X₅₁ = \((r^{2} + 2rR + s^{2})\) X₁ + \(6R(R - r)\) X₂ - \((r^{2} + 4rR - s^{2})\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines24 lines

{2, 262}, {4, 185}, {5, 52}, {6, 25}, {21, 970}, {22, 182}, {23, 575}, {24, 578}, {26, 569}, {31, 181}, {39, 237}, {44, 209}, {54, 288}, {107, 275}, {125, 132}, {129, 137}, {130, 138}, {199, 572}, {210, 374}, {216, 418}, {381, 568}, {397, 462}, {398, 463}, {573, 1011}

▸ Functional Relations27 relations

• X₅₁ = reflection of X₂₁₀ in X₃₇₅

• X₅₁ = isogonal conjugate of X₉₅

• X₅₁ = isotomic conjugate of X₃₄₃₈₄

• X₅₁ = complement of X₂₉₇₉

• X₅₁ = anticomplement of X₃₈₁₉

• X₅₁ = X_i-Ceva conjugate of X_j for these (i,j): (4, 53), (5, 216), (6, 217)

• X₅₁ = X₂-Ceva conjugate of complementary conjugate of X₃₄₈₄₅

• X₅₁ = X₂₁₇-cross conjugate of X₂₁₆

• X₅₁ = crosspoint of X_i and X_j for these (i,j): (4, 6), (5, 53)

• X₅₁ = crosssum of X_i and X_j for these (i,j): (2, 3), (6, 160), (54, 97)

• X₅₁ = crossdifference of every pair of points on line X₃₂₃X₄₀₁

• X₅₁ = inverse-in-orthosymmedial-circle of X₁₂₅

• X₅₁ = X₂ of tangential triangle of Johnson circumconic

• X₅₁ = trilinear pole of polar of X₂₇₆ wrt polar circle

• X₅₁ = pole wrt polar circle of trilinear polar of X₂₇₆ (line X₃₄₀X₅₂₀)

• X₅₁ = X₄₈-isoconjugate (polar conjugate) of X₂₇₆

• X₅₁ = X₉₂-isoconjugate of X₉₇

• X₅₁ = Zosma transform of X₉₂

• X₅₁ = intersection of tangents to Moses-Jerabek conic at X₆ and X₁₂₀₄

• X₅₁ = perspector of 1st & 2nd orthosymmedial triangles

• X₅₁ = bicentric sum of PU(157)

• X₅₁ = PU(157)-harmonic conjugate of X₆₄₇

• X₅₁ = perspector of orthic-of-orthocentroidal triangle and orthocentroidal-of-orthic triangle

• X₅₁ = centroid of reflection triangle of X₁₂₅

• X₅₁ = excentral-to-ABC functional image of X₂

• X₅₁ = centroid of anticomplementary circle intercepts of sidelines of ABC

• X₅₁ = {X₃₄₂₂₁,X₃₄₂₂₂}-harmonic conjugate of X₅₄₈₀

Notes

Let K_aK_bK_c be the anticomplementary triangle and let B_a and C_a be the orthogonal projections of K_b and K_c on BC, respectively. Define C_b and A_c cyclically, and define A_b and B_c cyclically. Then X₅₁ is the centroid of BaCaCbAbAcBc. (Randy Hutson, April 9, 2016)

Let L be the van Aubel line. Let U = X₆X₂₅, the isogonal conjugate of polar conjugate of L; let V = X₄X₅₁, the polar conjugate of the isogonal conjugate of L. Then X₅₁ = U∩V. (Randy Hutson, April 9, 2016)

Let H_aH_bH_c be the orthic triangle. Let O_a be the A-McCay circle of triangle AH_bH_c, and define O_b, O_c cyclically. X₅₁ is the radical center of circles O_a, O_b, O_c. (Randy Hutson, July 31 2018)

X₅₁ is the {X₅,X₁₄₃}-harmonic conjugate of X₅₂.