X₃₄₁₄

2nd KIEPERT INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₃₄₁₄ lies on the line at infinity.

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

X₃₄₁₄ (2nd 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, 1340}

• Related Centers: X₁₃₈₀ (2nd BROCARD-AXIS INTERCEPT OF CIRCUMCIRCLE), X₃₀ (EULER INFINITY POINT), X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈), X₂ (CENTROID), X₁₃₄₀ (INSIMILICENTER)

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

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

X₃₄₁₄ lies on the Kiepert hyperbola.

▸ Incident Central Lines33 lines

{2, 1340}, {3, 6177}, {4, 3557}, {5, 14633}, {30, 511}, {39, 14632}, {76, 13325}, {83, 14630}, {98, 1379}, {99, 1380}, {114, 2039}, {115, 2029}, {194, 3558}, {671, 6190}, {1312, 13870}, {1341, 7709}, {1348, 7694}, {1569, 2028}, {1670, 32481}, {1671, 32482}, {2037, 34454}, {2038, 3029}, {2482, 22245}, {2558, 10131}, {2559, 3406}, {5461, 22244}, {5466, 13722}, {5638, 9147}, {6039, 6054}, {6178, 13881}, {11257, 13326}, {14502, 14981}, {14631, 32467}

▸ Functional Relations13 relations

• X₃₄₁₄ = isogonal conjugate of X₁₃₈₀

• X₃₄₁₄ = isotomic conjugate of X₆₁₈₉

• X₃₄₁₄ = complementary conjugate of X₂₀₄₀

• X₃₄₁₄ = Thomson-isogonal conjugate of X₁₃₇₉

• X₃₄₁₄ = infinite point of minor axis of Steiner inellipse and Steiner circumellipse

• X₃₄₁₄ = bicentric difference of PU(i) for these i: 117, 119

• X₃₄₁₄ = ideal point of PU(i) for these i: 117, 119

• X₃₄₁₄ = trilinear pole of line {523, 13722}

• X₃₄₁₄ = crossdifference of every pair of points on line X₆X₅₆₃₈

• X₃₄₁₄ = X₂-Ceva conjugate of X₃₉₀₂₂

• X₃₄₁₄ = barycentric square root of X₃₉₀₂₂

• X₃₄₁₄ = perspector of hyperbola {{A, B, C,X₂,X₆₁₉₀,F₁”,F₂”}}, where F₁”, F₂” are the isotomic conjugates of PU(118) (the real foci of the Steiner inellipse)

• X₃₄₁₄ = {X₁₁₅,X₁₄₅₀₁}-harmonic conjugate of X₂₀₄₀