X₅₃

SYMMEDIAN POINT OF ORTHIC TRIANGLE

X₅₃ is the symmedian point of orthic triangle.

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

X₅₃ (SYMMEDIAN POINT 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–X₅₃, H_b–X₅₃, H_c–X₅₃ concurring at X₅₃

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

Barycentrics: \(a \tan A \cos(B - C) : b \tan B \cos(C - A) : c \tan C \cos(A - B)\)

Search Value (6, 9, 13): \(-0.134512425200\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines17 lines

{4, 6}, {5, 216}, {25, 157}, {30, 577}, {45, 281}, {115, 133}, {128, 139}, {137, 138}, {141, 264}, {232, 427}, {273, 1086}, {275, 288}, {311, 324}, {317, 524}, {318, 594}, {395, 472}, {396, 473}

▸ Functional Relations11 relations

• X₅₃ = isogonal conjugate of X₉₇

• X₅₃ = isotomic conjugate of X₃₄₃₈₆

• X₅₃ = anticomplement of X₃₄₈₂₈

• X₅₃ = X_i-Ceva conjugate of X_j for these (i,j): (4, 51), (324, 5)

• X₅₃ = X₅₁-cross conjugate of X₅

• X₅₃ = crosssum of X₃ and X₅₇₇

• X₅₃ = Kosnita(X₄,X₆) point

• X₅₃ = trilinear pole of line X₁₂₀₇₇X_15451 (the polar of X₉₅ wrt polar circle)

• X₅₃ = pole wrt polar circle of trilinear polar of X₉₅ (line X₃₂₃X₄₀₁)

• X₅₃ = polar conjugate of X₉₅

• X₅₃ = excentral-to-ABC functional image of X₆

Notes

Let Eu_aEu_bEu_c be the Euler triangle. Let L_A be the trilinear polar of Eu_a, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₅₃. (Randy Hutson, June 7, 2019)

X₅₃ is the {X₄,X₃₉₃}-harmonic conjugate of X₆.