X₇₂

ISOGONAL CONJUGATE OF X₂₈

X₇₂ is the isogonal conjugate of X₂₈.

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

X₇₂ (ISOGONAL CONJUGATE OF X₂₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 6}, {4, 8}

• Related Centers: X₂₈ (CEVAPOINT OF X₁₉ AND X₂₅), X₁ (INCENTER), X₆ (SYMMEDIAN POINT), X₄ (ORTHOCENTER), X₈ (NAGEL POINT)

Trilinears: \((b + c) \cot A : (c + a) \cot B : (a + b) \cot C\)

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

Search Value (6, 9, 13): \(5.668071554504\)

▸ Alternate Coordinates1 form

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

▸ Linear Combinations1 combination

• X₇₂ = \((r + 2R)\) X₁ - \(3R\) X₂ - \(r\) X₃ (Peter Moses, April 2, 2013)

X₇₂ lies on the Mandart hyperbola.

▸ Incident Central Lines34 lines

{1, 6}, {2, 942}, {3, 63}, {4, 8}, {5, 908}, {7, 443}, {10, 12}, {20, 144}, {21, 943}, {31, 976}, {35, 191}, {40, 64}, {43, 986}, {54, 1006}, {56, 997}, {57, 474}, {69, 304}, {73, 201}, {74, 100}, {145, 452}, {171, 1046}, {185, 916}, {190, 1043}, {222, 1038}, {248, 293}, {290, 668}, {295, 337}, {306, 440}, {394, 1060}, {519, 950}, {672, 1009}, {894, 1010}, {940, 975}, {978, 982}

▸ Functional Relations24 relations

• X₇₂ = reflection of X_i in X_j for these (i,j): (1, 960), (65, 10), (1145, 14740), (3555, 1)

• X₇₂ = isogonal conjugate of X₂₈

• X₇₂ = isotomic conjugate of X₂₈₆

• X₇₂ = inverse-in-Fuhrmann circle of X₃₄₁₉

• X₇₂ = anticomplement of X₉₄₂

• X₇₂ = X_i-Ceva conjugate of X_j for these (i,j): (8, 10), (63, 71), (69, 306), (321, 37)

• X₇₂ = X_i-cross conjugate of X_j for these (i,j): (201, 10), (228, 37)

• X₇₂ = crosspoint of X_i and X_j for these (i,j): (8, 78), (63, 69), (306, 307)

• X₇₂ = crosssum of X_i and X_j for these (i,j): (19, 25), (34, 56)

• X₇₂ = crossdifference of every pair of points on line X₅₁₃X₁₄₃₀

• X₇₂ = X_i-beth conjugate of X_j for these (i,j): (8, 65), (72, 73), (78, 72), (100, 227), (644, 72)

• X₇₂ = trilinear pole of line X₆₄₇X₆₅₆

• X₇₂ = complement of X₃₈₆₈

• X₇₂ = X₁₄₉ of X₁-Brocard triangle

• X₇₂ = X₆-isoconjugate of X₂₇

• X₇₂ = X₇₅-isoconjugate of X₂₂₀₃

• X₇₂ = X₉₂-isoconjugate of X₁₃₃₃

• X₇₂ = inverse-in-Fuhrmann-circle of X₃₄₁₉

• X₇₂ = X₆₁₄₆-of-excentral-triangle

• X₇₂ = perspector of ABC and cross-triangle of ABC and 2nd extouch triangle

• X₇₂ = trilinear product of Jerabek hyperbola intercepts of Nagel line

• X₇₂ = excentral-to-ABC barycentric image of X₄

• X₇₂ = antipode of X₁₁₄₅ in the Mandart hyperbola

• X₇₂ = extouch-isogonal conjugate of X₅₆₈₇

Notes

Let I_a be the reflection of X₁ in the perpendicular bisector of BC, and define I_b and I_c cyclically. X₇₂ = X₁₁ of I_aI_bI_c. (Randy Hutson, September 14, 2016)

X₇₂ is the {X₁,X₉}-harmonic conjugate of X₄₀₅.

X₇₂ is the perspector of the 1st and 2nd extouch triangles. X₇₂ is also the orthocenter of the 2nd extouch triangle. (Randy Hutson, August 23, 2011)