X₁₆₉₈

INSIMILICENTER(BEVAN CIRCLE, NINE-POINTCIRCLE)

X₁₆₉₈ is the insimilicenter(Bevan circle, nine-pointcircle).

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

X₁₆₉₈ (INSIMILICENTER(BEVAN CIRCLE, NINE-POINTCIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 2}, {5, 40}

• Related Centers: X₁ (INCENTER), X₂ (CENTROID), X₅ (NINE-POINT CENTER), X₄₀ (BEVAN POINT)

Trilinears: \(2 \cos(B - C) + \cos A - \cos B - \cos C + 1 : :\)

Search Value (6, 9, 13): \(2.817180849095\)

▸ Alternate Coordinates3 forms

• Trilinears: \(bc(a + 2b + 2c) : ca(b + 2c + 2a) : ab(c + 2a + 2b)\) (M. Iliev, 5/13/2007)

• Trilinears: \(bc - rR : ca - rR : ab - rR\) (C. Lozada, 9/07/2013)

• Trilinears: \(r - 4 R \sin B \sin C : :\)

▸ Linear Combinations1 combination

• X₁₆₉₈ = X₁ - \(6\) X₂ = X₈ - \(6\) X₁₀

▸ Incident Central Lines40 lines

{1, 2}, {4, 165}, {5, 40}, {9, 46}, {11, 1697}, {12, 57}, {33, 451}, {35, 405}, {36, 474}, {58, 750}, {75, 1089}, {115, 1571}, {116, 1282}, {119, 1768}, {121, 1054}, {140, 355}, {171, 1724}, {210, 942}, {226, 1788}, {281, 1838}, {318, 1784}, {320, 1757}, {404, 993}, {406, 1861}, {443, 1478}, {485, 1703}, {486, 1702}, {515, 631}, {517, 1656}, {595, 748}, {632, 952}, {966, 1743}, {986, 1739}, {1348, 1705}, {1349, 1704}, {1506, 1572}, {1676, 1701}, {1677, 1700}, {2009, 2018}, {2010, 2017}

▸ Functional Relations10 relations

• X₁₆₉₈ = isotomic conjugate of X₃₀₅₉₈

• X₁₆₉₈ = {X₂,X₁₀}-harmonic conjugate of X₁

• X₁₆₉₈ = homothetic center of excentral and 4th Euler triangles

• X₁₆₉₈ = crossdifference of every pair of points on line X₆₄₉X₂₆₀₅

• X₁₆₉₈ = trilinear product of vertices of Aquila triangle

• X₁₆₉₈ = homothetic center of ABC and cross-triangle of ABC and Aquila triangle

• X₁₆₉₈ = {X₁,X₂}-harmonic conjugate of X₃₆₂₄

• X₁₆₉₈ = perspector of Gemini triangle 26 and cross-triangle of ABC and Gemini triangle 26

• X₁₆₉₈ = isogonal conjugate of isotomic conjugate of X₃₀₅₉₆

• X₁₆₉₈ = homothetic center of Ai (aka K798i) triangle and cross-triangle of Fuhrmann and Ai triangles

Notes

Let A'B'C' be the Aquila triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. Then A″B″C″ is homothetic to ABC at X₁₆₉₈ and to A'B'C' at X₁₀. (Randy Hutson, December 10, 2016)