X₁₂₅

CENTER OF JERABEK HYPERBOLA

X₁₂₅ = complement of X₁₁₀

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

X₁₂₅ (CENTER OF JERABEK HYPERBOLA) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 67}, {4, 74}

• Related Centers: X₆ (SYMMEDIAN POINT), X₆₇ (ISOGONAL CONJUGATE OF X₂₃), X₄ (ORTHOCENTER), X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT)

Trilinears: \(\cos A \sin^{2}(B - C) : \cos B \sin^{2}(C - A) : \cos C \sin^{2}(A - B)\)

Barycentrics: \((\sin 2A)[\sin(B - C)]^{2} : (\sin 2B)[\sin(C - A)]^{2} : (\sin 2C)[\sin(A - B)]^{2}\)

Search Value (6, 9, 13): \(3.490271701682\)

▸ Alternate Coordinates2 forms

• Trilinears: \((\sec A)(c \cos C - b \cos B)^{2} : (\sec B)(a \cos A - c \cos C)^{2} : (\sec C)(b \cos B - a \cos A)^{2}\)

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

X₁₂₅ lies on these curves: nine-point circle, Walsmith rectangular hyperbola, orthic inconic, symmedial circle, Johnson circumconic of the medial triangle, cevian circle of every point on the Jerabek hyperbola, and bicevian conic of X₂ and X₇₂. X₁₂₅ also lies.

▸ Incident Central Lines16 lines

{2, 98}, {3, 131}, {4, 74}, {5, 113}, {6, 67}, {51, 132}, {68, 1092}, {69, 895}, {115, 245}, {119, 442}, {122, 684}, {126, 141}, {128, 140}, {136, 338}, {381, 541}, {511, 858}

▸ Functional Relations43 relations

• X₁₂₅ = midpoint of X_i and X_j for these (i,j): (3, 265), (4, 74), (6, 67), (110, 3448)

• X₁₂₅ = reflection of X_i in X_j for these (i,j): (113, 5), (185, 974), (1495, 468), (1511, 140), (1539, 546)

• X₁₂₅ = isogonal conjugate of X₂₅₀

• X₁₂₅ = isotomic conjugate of X₁₈₀₂₀

• X₁₂₅ = inverse-in-Brocard-circle of X₁₈₄

• X₁₂₅ = complement of X₁₁₀

• X₁₂₅ = complementary conjugate of X₅₂₃

• X₁₂₅ = X_i-Ceva conjugate of X_j for these (i,j): (4, 523), (66, 512), (68, 520), (69, 525), (338, 115)

• X₁₂₅ = crosspoint of X_i and X_j for these (i,j): (4, 523), (69, 525), (338, 339)

• X₁₂₅ = crosssum of X_i and X_j for these (i,j): (3, 110), (25, 112), (162, 270), (1113, 1114)

• X₁₂₅ = crossdifference of every pair of points on the line X₁₁₀X₁₁₂

• X₁₂₅ = X₁₁₅-Hirst inverse of X₈₆₈

• X₁₂₅ = X₂-line conjugate of X₁₁₀

• X₁₂₅ = orthopole of the Euler line

• X₁₂₅ = perspector of orthic triangle and Schroeter triangle

• X₁₂₅ = X₁₁₀-of-medial triangle

• X₁₂₅ = X₁₀₀-of-orthic triangle, if ABC is acute

• X₁₂₅ = X₈₅₈-of-1st-Brocard triangle

• X₁₂₅ = anticenter of cyclic quadrilateral ABCX₇₄

• X₁₂₅ = Λ(X₂₃₀, X₂₃₁), wrt orthic triangle

• X₁₂₅ = anticomplement of X₅₉₇₂

• X₁₂₅ = pole of Fermat axis wrt Dao-Moses-Telv circle

• X₁₂₅ = orthic-isogonal conjugate of X₅₂₃

• X₁₂₅ = perspector of circumconic centered at X₆₄₇

• X₁₂₅ = center of circumconic that is locus of trilinear poles of lines passing through X₆₄₇

• X₁₂₅ = X₂-Ceva conjugate of X₆₄₇

• X₁₂₅ = trilinear pole wrt orthic triangle of van Aubel line

• X₁₂₅ = inverse-in-polar-circle of X₁₀₇

• X₁₂₅ = inverse-in-{circumcircle, nine-point circle}-inverter of X₉₈

• X₁₂₅ = inverse-in-orthosymmedial-circle of X₅₁

• X₁₂₅ = centroid of (degenerate) pedal triangle of X₇₄

• X₁₂₅ = X_i-isoconjugate of X_j for these {i,j}: {4, 1101}, {92, 249}

• X₁₂₅ = inverse-in-Hutson-Parry-circle of X₈₆₈

• X₁₂₅ = {X₁₃₆₃₆,X₁₃₇₂₂}-harmonic conjugate of X₈₆₈

• X₁₂₅ = crosssum of MacBeath circumconic intercepts of Brocard axis

• X₁₂₅ = excentral-to-ABC functional image of X₁₀₀

• X₁₂₅ = 1st-Brocard-isotomic conjugate of X₃₅₉₀₁

• X₁₂₅ = antipode of X₁₄₉₅ in Walsmith rectangular hyperbola

• X₁₂₅ = orthocenter of X₆X₁₁₃X₃₅₆₉

• X₁₂₅ = orthocenter of X₁₁₀X₁₂₅X₃₅₆₉

• X₁₂₅ = orthocenter of X₁₄₉₅X₃₅₆₉X₃₅₈₀

• X₁₂₅ = polar conjugate of X₂₃₅₈₂

• X₁₂₅ = Moses-radical-circle-inverse of X₃₈₉₇₄

Notes

Roland H. Eddy and R. Fritsch, “The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle,” Mathematics Magazine 67 (1994) 188-205.

X₁₂₅ is the pole of the Fermat axis with respect to the Dao-Moses-Telv circle. (Randy Hutson, December 14, 2014)

Let L_a, L_b, L_c be the lines through A, B, C, resp. parallel to the Euler line. Let M_a, M_b, M_c be the reflections of BC, CA, AB in L_a, L_b, L_c, resp. Let A' = M_b∩M_c, and define B', C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A'B'C' in the Euler line. The triangle A″B″C″ is homothetic to ABC, with center of homothety X₁₂₅; see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

Let H_aH_bH_c be the orthic triangle. The Euler lines of AH_bH_c, BH_cH_a, CH_aH_b concur in X₁₂₅. (Randy Hutson, March 25, 2016)