X₁₁₃

JERABEK ANTIPODE

X₁₁₃ = complementary conjugate of X₃₀

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

X₁₁₃ (JERABEK ANTIPODE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Related Centers: X₃ (CIRCUMCENTER)

Trilinears: \(\sin B \sin C [(\sin C)/(\cos C - 2 \cos A \cos B) + (\sin B)/(\cos B - 2 \cos A \cos C)] : :\)

Barycentrics: \((\sin C)/(\cos C - 2 \cos A \cos B) + (\sin B)/(\cos B - 2 \cos A \cos C) : :\)

Search Value (6, 9, 13): \(-2.384528218412\)

▸ Alternate Coordinates2 forms

• Barycentrics: \(b^{2}\)/(\(b^{2}\)\(S_{B}\) - \(2S_{A}\)\(S_{C})\) + \(c^{2}\)/(\(c^{2}\)\(S_{C}\) - \(2S_{A}\)\(S_{B})\) : : (Peter J. C. Moses, 3/2003)

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

X₁₁₃ lies on the bicevian conic of X₂ and X₁₁₀, the Walsmith rectangular hyperbola.

▸ Incident Central Lines11 lines

{2, 74}, {3, 122}, {4, 110}, {5, 125}, {6, 13}, {11, 942}, {52, 135}, {114, 690}, {123, 960}, {127, 141}, {137, 546}

▸ Functional Relations29 relations

• X₁₁₃ = midpoint of X_i and X_j for these (i,j): (4, 110), (74, 146), (265, 399), (1553, 3258)

• X₁₁₃ = reflection of X_i in X_j for these (i,j): (52, 1112), (125, 5), (32110, 468)

• X₁₁₃ = complementary conjugate of X₃₀

• X₁₁₃ = X_i-Ceva conjugate of X_j for these (i,j): (4, 30), (2, 3003)

• X₁₁₃ = crosspoint of X₄ and X₄₀₃

• X₁₁₃ = crossdifference of every pair of points on line X₅₂₆X₆₈₆

• X₁₁₃ = nine-point-circle-antipode of X₁₂₅

• X₁₁₃ = X₇₄-of-medial-triangle

• X₁₁₃ = X₁₀₄-of-orthic-triangle if ABC is acute

• X₁₁₃ = X₁₈₆-of-X₄-Brocard-triangle

• X₁₁₃ = center of rectangular circumhyperbola that passes through X₁₁₀

• X₁₁₃ = center of rectangular hyperbola {X₃,X₄,X₁₁₀,X₁₅₅,X₁₃₅₁,X₁₃₅₂,X(2574,X₂₅₇₅}}

• X₁₁₃ = perpsector of circumconic centered at X₃₀₀₃

• X₁₁₃ = inverse-in-polar-circle-of X₁₃₀₀

• X₁₁₃ = inverse-in-{circumcircle, nine-point circle}-inverter of X₁₃₀₂

• X₁₁₃ = anticenter of cyclic quadrilateral ABCX₁₁₀

• X₁₁₃ = Λ(X₂,X₃)-with-respect-to-orthic-triangle

• X₁₁₃ = barycentric product X₃₀*X₃₅₈₀

• X₁₁₃ = complement of X₇₄

• X₁₁₃ = reflection of X₃ in X₅₉₇₂

• X₁₁₃ = antipode of X₃ in the bicevian conic of X₂ and X₁₁₀

• X₁₁₃ = antipode of X₅₂ in the Hatzipolakis-Lozada hyperbola

• X₁₁₃ = orthopole of line X₃X₅₂₃

• X₁₁₃ = perspector of Ehrmann mid-triangle and orthic triangle

• X₁₁₃ = excentral-to-ABC functional image of X₁₀₄

• X₁₁₃ = antipode of X₃₂₁₁₀ in Walsmith rectangular hyperbola

• X₁₁₃ = orthocenter of X₆X₁₂₅X₃₅₆₉

• X₁₁₃ = crosssum of circumcircle intercepts of line PU(161) (line X₃X₅₂₃)

• X₁₁₃ = homothetic center of ABC and X₃₀-Fuhrmann triangle

Notes

Let H_aH_bH_c be the orthic triangle. Let L be the line through H_a parallel to the Euler line, and define M and N cyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L', M', N' concur in X₁₁₃. (Randy Hutson, August 26, 2014)

Let H_aH_bH_c be the orthic triangle. Let M_A be the reflection of the orthic axis in line H_bH_c, and define Let M_B and Let M_C cyclically. Let A″ = Let M_B∩M_C, and define B″ and C″ cyclically. The lines H_aA″, H_bB″ H_cC″ concur in X₁₁₃. (Randy Hutson, August 26, 2014)

Let H_aH_bH_c be the orthic triangle. Let N_A be the orthic axis of AH_bH_c, and define N_B and N_C cyclically. Let A″ = N_B∩N_C, B″ = N_C∩A_C, C″ = N_A∩B_C. Then triangle A″B″C″ is inversely similar to ABC, with similitude center X₆, and the lines H_aA″, H_bB″, H_cC″ concur in X₁₁₃. Also, X₁₁₃ = X₃-of-A″B″C″. (Randy Hutson, August 26, 2014)