X₁₆₉₉

EXSIMILICENTER(BEVAN CIRCLE, NINE-POINT CIRCLE)

X₁₆₉₉ is the exsimilicenter(Bevan circle, nine-point circle).

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

X₁₆₉₉ (EXSIMILICENTER(BEVAN CIRCLE, NINE-POINT CIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {165, 2}

• Related Centers: X₁₆₅ (CENTROID OF THE EXCENTRAL TRIANGLE), X₂ (CENTROID)

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

Barycentrics: \(a^{3} + a(b - c)^{2} - 2 (b - c)^{2} (b + c) : :\)

Search Value (6, 9, 13): \(-3.220976770799\)

▸ Alternate Coordinates1 form

• Trilinears: \(r + 4 R \cos B \cos C : :\)

▸ Linear Combinations1 combination

• X₁₆₉₉ = X₁ + \(2\) X₄

▸ Incident Central Lines32 lines

{1, 4}, {2, 165}, {5, 40}, {10, 962}, {11, 57}, {12, 1697}, {20, 1125}, {36, 1012}, {55, 1538}, {79, 84}, {80, 1537}, {115, 1572}, {118, 1282}, {200, 908}, {210, 381}, {238, 1754}, {354, 971}, {355, 546}, {382, 1385}, {485, 1702}, {486, 1703}, {499, 1770}, {610, 1839}, {614, 990}, {1329, 1706}, {1348, 1704}, {1506, 1571}, {1676, 1700}, {1677, 1701}, {1730, 1985}, {2009, 2017}, {2010, 2018}

▸ Functional Relations10 relations

• X₁₆₉₉ = reflection of X₁₆₅ in X₂

• X₁₆₉₉ = crosspoint of X₉₂ and X₁₀₈₈

• X₁₆₉₉ = crosssum of X₄₈ and X₁₂₅₃

• X₁₆₉₉ = centroid of the six touchpoints of the Johnson circles and the sidelines of the inner Johnson triangle

• X₁₆₉₉ = homothetic center of excentral and 3rd Euler triangles

• X₁₆₉₉ = centroid of triangle formed by reflecting excenters in corresponding vertex of ABC

• X₁₆₉₉ = centroid of anticevian triangle, wrt intouch triangle, of X₁

• X₁₆₉₉ = homothetic center of circumcevian triangle of X₃ and cross-triangle of Aquila and anti-Aquila triangles

• X₁₆₉₉ = centroid of Garcia reflection triangle (aka Gemini triangle 8)

• X₁₆₉₉ = {X₁,X₄}-harmonic conjugate of X₅₆₉₁

Notes

Let A' be the pole of the Gergonne line wrt the circle with BC as diameter, and define B', C' cyclically. X₁₆₉₉ is the centroid of A'B'C'. (Randy Hutson, June 27, 2018)

Let A' be the orthocenter of BCX₁, and define B' and C' cyclically. A'B'C' is also the anticevian triangle, wrt intouch triangle, of X₁, and X₁₆₉₉ is the centroid of A'B'C'; see also X₃₆₈₀. (Randy Hutson, June 27, 2018)

X₁₆₉₉ is the centroid of triangle formed by reflecting vertices of 1st circumperp triangle in corresponding side of ABC. (If 2nd circumperp triangle is substituted, for 1st, the resulting triangle is the Fuhrmann triangle.) (Randy Hutson, June 27, 2018)

Let H_aH_bH_c be the orthic triangle. Let A″ be the A-excenter of AH_bH_c, and define B″ and C″ cyclically. The centroid of triangle A″B″C″ is X₁₆₉₉. (Randy Hutson, July 31 2018)