X₅₂₅

ISOGONAL CONJUGATE OF X₁₁₂

As the isogonal conjugate of a point on the circumcircle, X₅₂₅ lies on the line at infinity.

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

X₅₂₅ (ISOGONAL CONJUGATE OF X₁₁₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {99, 249}, {30, 511}

• Related Centers: X₁₁₂ (Ψ(ORTHOCENTER, SYMMEDIAN POINT)), X₉₉ (STEINER POINT), X₂₄₉ (ISOGONAL CONJUGATE OF X₁₁₅), X₃₀ (EULER INFINITY POINT), X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈)

Trilinears: \((b \cos B - c \cos C)/a : (c \cos C - a \cos A)/b : (a \cos A - b \cos B)/c\)

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

▸ Alternate Coordinates5 forms

• Trilinears: \(\cot A \sin(B - C) : :\)

• Barycentrics: \(\sin 2B - \sin 2C : :\)

• Barycentrics: \(\tan B - \tan C : :\)

• Barycentrics: \(\tan B \cos 2B - \tan C \cos 2C : :\)

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

▸ Incident Central Lines29 lines

{2, 1640}, {3, 878}, {4, 2435}, {30, 511}, {74, 2697}, {98, 2710}, {99, 249}, {100, 2722}, {103, 2741}, {104, 2747}, {105, 2754}, {106, 2760}, {107, 2764}, {109, 2769}, {110, 935}, {112, 2867}, {127, 1562}, {297, 850}, {323, 401}, {339, 3269}, {441, 647}, {669, 2528}, {1073, 2416}, {1636, 3268}, {1975, 2422}, {2474, 2514}, {2485, 2506}, {2513, 2531}, {2632, 2968}

▸ Functional Relations21 relations

• X₅₂₅ = isogonal conjugate of X₁₁₂

• X₅₂₅ = isotomic conjugate of X₆₄₈

• X₅₂₅ = complementary conjugate of X₁₂₇

• X₅₂₅ = X_i-Ceva conjugate of X_j for these (i,j): (2, 15526), (4, 127), (69, 125), (76, 339), (99, 3), (110, 141), (190, 440), (253, 122)

• X₅₂₅ = X_i-cross conjugate of X_j for these (i,j): (115, 68), (122, 253), (125, 69)

• X₅₂₅ = crosspoint of X₇₆ and X₉₉

• X₅₂₅ = crosssum of X_i and X_j for these (i,j): (6, 647), (32, 512), (427, 523)

• X₅₂₅ = crossdifference of every pair of points on line X₆X₂₅

• X₅₂₅ = perspector of hyperbola {A, B, C,X₂,X₆₉}}

• X₅₂₅ = intersection of trilinear polars of X₂ and X₆₉

• X₅₂₅ = trilinear pole of line X₁₂₂X₁₂₅

• X₅₂₅ = pole wrt polar circle of trilinear polar of X₁₀₇ (line X₄X₆)

• X₅₂₅ = X₄₈-isoconjugate (polar conjugate) of X₁₀₇

• X₅₂₅ = X₆-isoconjugate of X₁₆₂

• X₅₂₅ = X₉₂-isoconjugate of X₁₅₇₆

• X₅₂₅ = intersection of tangents to Steiner inellipse at X₂₄₅₄ and X₂₄₅₅

• X₅₂₅ = crosspoint wrt medial triangle of X₂₄₅₄ and X₂₄₅₅

• X₅₂₅ = Thomson-isogonal conjugate of X₁₂₉₇

• X₅₂₅ = Lucas-isogonal conjugate of X₁₂₉₇

• X₅₂₅ = crosssum of Jerabek hyperbola intercepts of Lemoine axis

• X₅₂₅ = barycentric product of Jerabek hyperbola intercepts of de Longchamps line

Notes

X₅₂₅ is the barycentric multiplier for the Jerabek hyperbola. (The barycentric product of X₅₂₅ and the circumcircle is the Jerabek hyperbola.) (Randy Hutson, August 19, 2019)