X₁₁₄

KIEPERT ANTIPODE

X₁₁₄ is the QA-P30 center (Reflection of QA-P₂ in QA-P11) of quadrangle ABCX₂ (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/58-qa-p30.html)

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

X₁₁₄ (KIEPERT 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: \(bc[b \sec(B + \omega ) + c \sec(C + \omega )] : :\)

Barycentrics: \(b \sec(B + \omega ) + c \sec(C + \omega ) : :\)

Search Value (6, 9, 13): \(-1.118944978639\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\cos(B - C) \cos 2\omega - \sin \omega \sin(A + \omega )\) (Peter J. C. Moses, 9/12/03)

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

X₁₁₄ lies on the bicevian conic of X₂ and X₉₉.

▸ Incident Central Lines11 lines

{2, 98}, {3, 127}, {4, 99}, {5, 39}, {25, 135}, {52, 211}, {113, 690}, {132, 684}, {136, 427}, {325, 511}, {381, 543}

▸ Functional Relations24 relations

• X₁₁₄ = midpoint of X_i and X_j for these (i,j): (4, 99), (98, 147)

• X₁₁₄ = reflection of X_i in X_j for these (i,j): (3, 620), (115, 5)

• X₁₁₄ = isogonal conjugate of X₂₀₆₅

• X₁₁₄ = complement of X₉₈

• X₁₁₄ = complementary conjugate of X₅₁₁

• X₁₁₄ = X_i-Ceva conjugate of X_j for these (i,j): (2, 230), (4, 511)

• X₁₁₄ = crosspoint of X₂ and X₃₂₅

• X₁₁₄ = orthojoin of X₂₃₀

• X₁₁₄ = Moses-radical-circle-inverse of X₃₈₉₇₅

• X₁₁₄ = nine-point-circle-antipode of X₁₁₅

• X₁₁₄ = X₉₈-of-medial triangle

• X₁₁₄ = X₁₀₃-of-orthic triangle if ABC is acute

• X₁₁₄ = perspector of circumconic centered at X₂₃₀

• X₁₁₄ = center of circumconic that is locus of trilinear poles of lines passing through X₂₃₀

• X₁₁₄ = center of rectangular hyperbola {X₄,X₇₆,X₉₉,X₃₇₆,X₄₈₇,X₄₈₈}}

• X₁₁₄ = X₁₅₁₃-of-1st-Brocard-triangle

• X₁₁₄ = inverse-in-polar-circle of X₃₅₆₃

• X₁₁₄ = inverse-in-{circumcircle, nine-point circle}-inverter of X₁₁₀

• X₁₁₄ = center of inverse-in-{circumcircle, nine-point circle}-inverter of Brocard circle

• X₁₁₄ = X₅-of-1st-anti-Brocard-triangle

• X₁₁₄ = anticenter of cyclic quadrilateral ABCX₉₉

• X₁₁₄ = Λ(X₃, X₆), wrt orthic triangle

• X₁₁₄ = orthocenter of X₉₈X₁₁₅X₃₁₉₅₃

• X₁₁₄ = centroid of mid-triangle of X₁₅- and X₁₆-Fuhrmann triangles

Notes

Let H_aH_bH_c be the orthic triangle. Let L_a be the Lemoine axis of triangle AH_bH_c, and define L_b and L_c cyclically. Let A″ = L_b∩L_c, and define B″ and C″ cyclically. Triangle A″B″C″ is inversely similar to ABC, with similicenter X₂. The lines H_aA″, H_bB″, H_cC″ concur in X₁₁₄, which is X₃-of-A″B″C″.

Let O_A be the circle centered at the A-vertex of the obverse triangle of X₆₉ and passing through A; define O_B and O_C cyclically. X₁₁₄ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)