X₁₆₅

CENTROID OF THE EXCENTRAL TRIANGLE

X₁₆₅ is the {X₃,X₄₀}-harmonic conjugate of X₁. For a list of harmonic conjugates of X₁₆₅, click Tables at the top of this page.

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

X₁₆₅ (CENTROID OF THE EXCENTRAL TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Excentral Triangle △J_aJ_bJ_c — vertices J_a, J_b, J_c (Anticevian triangle of Incenter X₁ (the three excenters))

• Concurrence Lines: J_a–X₁₆₅, J_b–X₁₆₅, J_c–X₁₆₅ concurring at X₁₆₅

Trilinears: \(\tan(B/2) + \tan(C/2) - \tan(A/2) : \tan(C/2) + \tan(A/2) - \tan(B/2) : \tan(A/2) + \tan(B/2) - \tan(C/2)\)

Barycentrics: \(a[\tan(B/2) + \tan(C/2) - \tan(A/2)] : :\)

Search Value (6, 9, 13): \(8.479714355776\)

▸ Alternate Coordinates2 forms

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

• Trilinears: \((r/R) - 4 \cos A : :\)

▸ Linear Combinations1 combination

• X₁₆₅ = X₁ - \(4\) X₃

▸ Incident Central Lines45 lines

{1, 3}, {2, 516}, {4, 1698}, {9, 910}, {10, 20}, {32, 1571}, {42, 991}, {43, 573}, {63, 100}, {71, 610}, {105, 1054}, {108, 1767}, {109, 212}, {164, 167}, {166, 168}, {191, 1079}, {210, 971}, {218, 1190}, {220, 1615}, {227, 1394}, {255, 1103}, {269, 1253}, {355, 550}, {371, 1703}, {372, 1702}, {376, 515}, {380, 579}, {386, 1695}, {411, 936}, {479, 1323}, {498, 1770}, {572, 1051}, {574, 1572}, {580, 601}, {612, 990}, {614, 902}, {631, 946}, {750, 968}, {846, 1719}, {950, 1788}, {958, 1706}, {962, 1125}, {1011, 1730}, {1342, 1701}, {1343, 1700}

▸ Functional Relations17 relations

• X₁₆₅ = isogonal conjugate of X₃₀₆₂

• X₁₆₅ = X₉-Ceva conjugate of X₁

• X₁₆₅ = anticomplement of X₃₈₁₇

• X₁₆₅ = X₂-of-1st-circumperp-triangle

• X₁₆₅ = homothetic center of ABC and orthic triangle of 1st circumperp triangle

• X₁₆₅ = homothetic center of excentral triangle and medial triangle of 1st circumperp triangle

• X₁₆₅ = excentral isogonal conjugate of X₉

• X₁₆₅ = excentral isotomic conjugate of X₁₆₅

• X₁₆₅ = excentral polar conjugate of X₁

• X₁₆₅ = Thomson-isogonal conjugate of X₁

• X₁₆₅ = reflection of X₁₆₉₉ in X₂

• X₁₆₅ = cyclocevian conjugate of X₁ wrt anticevian triangle of X₁

• X₁₆₅ = X_i-beth conjugate of X_j for these (i,j): (100, 165), (643, 200)

• X₁₆₅ = X_i-aleph conjugate of X_j for these (i,j): (2, 169), (9, 165), (21, 572), (100, 101), (188, 9), (259, 43), (365, 978), (366, 57), (650, 1053)

• X₁₆₅ = centroid of the triangle with vertices X₁, X₈, X₂₀

• X₁₆₅ = centroid of the triangle with vertices X₄, X₂₀, X₄₀

• X₁₆₅ = ABC-to-excentral barycentric image of X₂

Notes

If DEF is the pedal triangle of X₁₆₅, then |AE| + |AF| = |BF| + |BD| = |CD| + |CE|. (Seiichi Kirikami, October 8, 2010.)

Let A'B'C' be the anticevian triangle, wrt intouch triangle, of X₁. Let A″ be the reflection of A' in A, and define B' and C' cyclically. The centroid of A″B″C″ is X₁₆₅. (Randy Hutson, December 2, 2017)

Let J_aJ_bJ_c be the excentral triangle. Let A″ be the symmedian point of triangle J_aBC, and define B″ and C″ cyclically. The lines J_aA″, J_bB″, J_cC″ concur in X₁₆₅. (Randy Hutson, July 31 2018)