X₃₀₆₀

INTERSECTION OF LINES X₂X_51 AND X₄X_52

X₃₀₆₀ = reflection of X₂ in X₅₁

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

X₃₀₆₀ (INTERSECTION OF LINES X₂X_51 AND X₄X_52) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 51}

• Related Centers: X₂ (CENTROID), X₅₁ (CENTROID OF ORTHIC TRIANGLE)

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

Search Value (6, 9, 13): \(-2.388117627074\)

▸ Alternate Coordinates1 form

• Trilinears: \(\sin A (\sin 2B + \sin 2C) - \sin B \sin C : :\)

▸ Incident Central Lines16 lines

{2, 51}, {3, 143}, {4, 52}, {6, 22}, {20, 389}, {23, 184}, {25, 110}, {26, 54}, {143, 1173}, {156, 195}, {371, 588}, {372, 589}, {394, 1995}, {648, 1629}, {1853, 2781}, {1992, 2393}

▸ Functional Relations10 relations

• X₃₀₆₀ = centroid of the intersections, other than A, B, C, of circles {{X₄,B,C}}, {{X₄,C,A}}, {{X₄,A,B}} and lines BC, CA, AB. (Randy Hutson, March 25, 2016)

• X₃₀₆₀ = reflection of X₂ in X₅₁

• X₃₀₆₀ = anticomplement of X₃₉₁₇

• X₃₀₆₀ = centroid of reflection triangle

• X₃₀₆₀ = {X₂,X₅₁}-harmonic conjugate of X₅₆₄₀

• X₃₀₆₀ = X₁₆₉₉-of-orthic-triangle, if ABC is acute; see X₃₅₆₇

• X₃₀₆₀ = homothetic center of orthic-of-orthic triangle and circumorthic-of-circumorthic triangle

• X₃₀₆₀ = homothetic center of orthocentroidal triangle and X₂-adjunct anti-altimedial triangle

• X₃₀₆₀ = X₃₄₄₈-of-orthocentroidal-triangle

• X₃₀₆₀ = exsimilicenter of the circumcircle and the nine-point circle of the orthic triangle. (Peter Moses, July 1, 2009)

Notes

X₃₀₆₀ is the second of three Spanish Points developed by Antreas P. Hatzipolakis and Javier Garcia Capitan in 2009; see X₃₅₆₇.

Let A' be the trilinear pole of the perpendicular bisector of BC, and define B' and C' cyclically. A'B'C' is also the anticomplement of the anticomplement of the midheight triangle. X₃₀₆₀ = X₂-of-A'B'C'. (Randy Hutson, January 29, 2018)