X₃₄₁₃

1st KIEPERT INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₃₄₁₃ lies on the line at infinity.

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

X₃₄₁₃ (1st KIEPERT INFINITY POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {30, 511}, {2, 1341}

• Related Centers: X₁₃₇₉ (1st BROCARD-AXIS INTERCEPT OF CIRCUMCIRCLE), X₃₀ (EULER INFINITY POINT), X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈), X₂ (CENTROID), X₁₃₄₁ (EXSIMILICENTER)

Trilinears: \(1/(e \cos A - \cos(A + \omega )) : :\)

Barycentrics: \(1/(S_{A}^{2} - S_{B} S_{C} - S_{A} \sqrt{-S^{2} + S_{A}^{2} + S_{B}^{2} + S_{C}^{2}})) : :\)

▸ Alternate Coordinates1 form

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

X₃₄₁₃ lies on the Kiepert hyperbola.

▸ Incident Central Lines39 lines

{2, 1341}, {3, 6178}, {4, 3558}, {5, 14632}, {30, 511}, {39, 14633}, {76, 13326}, {83, 14631}, {98, 1380}, {99, 1379}, {100, 36735}, {104, 36736}, {114, 2040}, {115, 2028}, {194, 3557}, {671, 6189}, {1313, 13870}, {1340, 7709}, {1349, 7694}, {1569, 2029}, {1670, 32482}, {1671, 32481}, {2037, 3029}, {2038, 34454}, {2482, 22244}, {2558, 3406}, {2559, 10131}, {5461, 22245}, {5466, 13636}, {5639, 9147}, {6040, 6054}, {6177, 13881}, {11257, 13325}, {12188, 38597}, {13188, 38596}, {14501, 14981}, {14630, 32467}, {39103, 39108}, {39105, 39107}

▸ Functional Relations13 relations

• X₃₄₁₃ = isogonal conjugate of X₁₃₇₉

• X₃₄₁₃ = isotomic conjugate of X₆₁₉₀

• X₃₄₁₃ = complementary conjugate of X₂₀₃₉

• X₃₄₁₃ = Thomson-isogonal conjugate of X₁₃₈₀

• X₃₄₁₃ = infinite point of major axis of Steiner inellipse and Steiner circumellipse

• X₃₄₁₃ = bicentric difference of PU(i) for these i: 116, 118

• X₃₄₁₃ = ideal point of PU(i) for these i: 116, 118

• X₃₄₁₃ = trilinear pole of line {523, 13636}

• X₃₄₁₃ = crossdifference of every pair of points on line X₆X₅₆₃₉

• X₃₄₁₃ = X₂-Ceva conjugate of X₃₉₀₂₃

• X₃₄₁₃ = barycentric square root of X₃₉₀₂₃

• X₃₄₁₃ = perspector of hyperbola {{A, B, C,X₂,X₆₁₈₉,F₁”,F₂”}}, where F₁”, F₂” are the isotomic conjugates of PU(119) (the imaginary foci of the Steiner inellipse)

• X₃₄₁₃ = {X₁₁₅,X₁₄₅₀₂}-harmonic conjugate of X₂₀₃₉

Notes

The asymptotes of the Kiepert hyperbola meet the infinity line in X₃₄₁₃ and X₃₄₁₄. See the note at X₂₅₇₄.