X₁₂₇

X₁₁₂-OF-MEDIAL-TRIANGLE

X₁₂₇ = reflection of X₁₃₂ in X₅

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

X₁₂₇ (X₁₁₂-OF-MEDIAL-TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {132, 5}

• Related Centers: X₁₃₂ (∩X₄X₃₂), X₅ (NINE-POINT CENTER)

Trilinears: \(bc(\sin 2B - \sin 2C)[(b^{2} - c^{2})\sin 2A - b^{2}\sin 2B + c^{2}\sin 2C] : :\)

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

Search Value (6, 9, 13): \(4.044284549477\)

▸ Alternate Coordinates1 form

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

X₁₂₇ lies on the nine-point circle, the cevian circle of X₂₂.

▸ Incident Central Lines7 lines

{2, 112}, {3, 114}, {5, 132}, {113, 141}, {115, 338}, {133, 381}, {125, 140}

▸ Functional Relations17 relations

• X₁₂₇ = reflection of X₁₃₂ in X₅

• X₁₂₇ = isogonal conjugate of X₁₅₃₈₈

• X₁₂₇ = isotomic conjugate of isogonal conjugate of X₃₈₃₅₆

• X₁₂₇ = complement of X₁₁₂

• X₁₂₇ = anticomplementary conjugate of X₅₂₅

• X₁₂₇ = X₄-Ceva conjugate of X₅₂₅

• X₁₂₇ = X₁₂₉₂-of-orthic-triangle if ABC is acute

• X₁₂₇ = perspector of circumconic centered at X₂₄₈₅

• X₁₂₇ = center of circumconic that is locus of trilinear poles of lines passing through X₂₄₈₅

• X₁₂₇ = X₂-Ceva conjugate of X₂₄₈₅

• X₁₂₇ = inverse-in-polar-circle of X₁₂₈₉

• X₁₂₇ = inverse-in-orthoptic-circle-of-Steiner-inellipse of X₂₃₇₃

• X₁₂₇ = X₁₁₂-of-medial triangle

• X₁₂₇ = center of the rectangular hyperbola that passes through A, B, C, and X₂₂

• X₁₂₇ = crosssum of circumcircle intercepts of line X₃X₆₆

• X₁₂₇ = orthopole of line X₃X₆₆

• X₁₂₇ = Kirikami-six-circles image of X₂₂

Notes

Let H_aH_bH_c be the orthic triangle. Let L_a be the van Aubel line 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. The lines H_aA″, H_bB″, H_cC″ concur in X₁₂₇, which is X₃-of-A″B″C″. (Randy Hutson, July 31 2018)