X₁₆₃₇

TRIPOLAR CENTROID OF X₄

X₁₆₃₇ is the tripolar centroid of X₄.

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

X₁₆₃₇ (TRIPOLAR CENTROID OF X₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 3268}

• Related Centers: X₂ (CENTROID), X₃₂₆₈ (ISOTOMIC CONJUGATE OF X₄₇₆)

Barycentrics: \(\tan A (\tan B - \tan C)(\tan B + \tan C - 2 \tan A) : :\)

Search Value (6, 9, 13): \(-3.108673947477\)

▸ Alternate Coordinates2 forms

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

• Barycentrics: \((\cot B - \cot C)(\tan B + \tan C - 2 \tan A) : :\)

▸ Incident Central Lines8 lines

{2, 3268}, {98, 111}, {107, 112}, {115, 125}, {132, 1560}, {230, 231}, {1499, 1514}, {1636, 1651}

▸ Functional Relations19 relations

• X₁₆₃₇ = complement of X₃₂₆₈

• X₁₆₃₇ = X₁₉₈₉-Ceva conjugate of X₁₁₅

• X₁₆₃₇ = crosspoint of X₂ and X₄₇₆

• X₁₆₃₇ = crosssum of X_i and X_j for these (i,j): (3, 1636), (6, 526)

• X₁₆₃₇ = crossdifference of every pair of points on line X₃X₇₄

• X₁₆₃₇ = PU(4)-harmonic conjugate of X₆₁₀₃

• X₁₆₃₇ = midpoint of circumcenters of X₁₃X₁₄X₁₅ and X₁₃X₁₄X₁₆

• X₁₆₃₇ = perspector of circumconic centered at X₃₂₅₈

• X₁₆₃₇ = center of circumconic that is locus of trilinear poles of lines passing through X₃₂₅₈

• X₁₆₃₇ = X₂-Ceva conjugate of X₃₂₅₈

• X₁₆₃₇ = centroid of orthic axis intercepts with sidelines of ABC

• X₁₆₃₇ = tripolar centroid of X₄ wrt orthic triangle

• X₁₆₃₇ = center of Dao-Moses-Telv circle

• X₁₆₃₇ = X₁₁₅ of 2nd Parry triangle

• X₁₆₃₇ = insimilicenter of circles {{X₉₈,X₁₀₇,X₁₂₅,X₁₃₂}} and {{X₆,X₁₁₁,X₁₁₂,X₁₁₅,X₁₈₇,X₁₅₆₀}}; the exsimilicenter is X₆₁₀₃

• X₁₆₃₇ = radical center of orthocentroidal circles of ABC, orthocentroidal triangle, anti-orthocentroidal triangle

• X₁₆₃₇ = inverse-in-Hutson-Parry-circle of X₁₆₄₀

• X₁₆₃₇ = {X₁₃₆₃₆,X₁₃₇₂₂}-harmonic conjugate of X₁₆₄₀

• X₁₆₃₇ = centroid of (degenerate) side triangle of 3rd and 4th isodynamic-Dao triangles

Notes

Let A'B'C' and A″B″C″ be the cevian and anticevian triangles of X₄, resp. Let O_A be the cevian circle of A″. Let A* be the intersection, other than A', of O_A and line BC. Define B* and C* cyclically. X₁₆₃₇ is the centroid of triangle A*B*C*. (Randy Hutson, August 28, 2020)