X₁₁₁₅

EXTERIOR-ANGLE CURVATURE CENTROID

X₁₁₁₅ is the center of mass of a point-mass system obtained by placing at vertex :g:`A <A*>` a mass equal to the magnitude of the exterior angle (that’s π - :g:`A <A*>`) at :g:`A <A*>`, and cyclically for :g:`B <B*>` and :g:`C <C*>`. ( Peter Scales , Hyacinthos #5528, 5/22/02) This description is also found in Honsberger , Episodes in Nineteenth and Twentieth Century Eulidean Geometry , p. 120, where it is mistakenly attributed to the Steiner point, X₉₉.

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

X₁₁₁₅ (EXTERIOR-ANGLE CURVATURE CENTROID) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 360}

• Related Centers: X₂ (CENTROID), X₃₆₀ (HOFSTADTER ZERO POINT)

Trilinears: \((\pi - A)/a : (\pi - B)/b : (\pi - C)/c\)

Barycentrics: \(\pi - A : \pi - B : \pi - C\)

Search Value (6, 9, 13): \(3.421226357111\)

▸ Linear Combinations1 combination

• X₁₁₁₅ = \(9\) X₂ - \(5\) X₁₈₂₉₅, \(3\) X₃₆₀ - \(5\) X₁₈₂₉₅, \(6\) X₁₈₂₉₄ - \(5\) X₁₈₂₉₅

▸ Incident Central Lines1 line

{2, 360}

▸ Functional Relations10 relations

• X₁₁₁₅ = reflection of X₃₆₀ in X₁₈₂₉₄

• X₁₁₁₅ = isogonal conjugate of X₇₀₂₁

• X₁₁₁₅ = complement of X₃₆₀

• X₁₁₁₅ = anticomplement of X₁₈₂₉₄

• X₁₁₁₅ = complement of the isogonal conjugate of X₃₅₉

• X₁₁₁₅ = X_i-complementary conjugate of X_j for these (i,j): {359, 10}, {1077, 141}

• X₁₁₁₅ = X_i-isoconjugate of X_j for these (i,j): {1, 7021}, {6, 7041}

• X₁₁₁₅ = barycentric product X₇₅*X₇₀₃₉

• X₁₁₁₅ = barycentric quotient X_i/X_j for these {i,j}: {1, 7041}, {6, 7021}, {7039, 1}, {7041, 7036}, {7044, 7039}

• X₁₁₁₅ = {X₂,X₃₆₀}-harmonic conjugate of X₁₈₂₉₄

Notes

X₁₁₁₅ should not be confused with Jakob Steiner’s actual “Curvature Centroid” (discovered in 1825), applicable to all polygons, obtained as the weighted sum of vertices with weights determined by sines of doubled vertex angles [1]. Remarkably, the pedal polygon of a polygon with respect to this point has extremal area: it is minimal (resp. maximal) if the sign of the sum of sines of double angles is negative (resp. positive) [1] . For a triangle, this point is the circumcenter X₃ and the extremal pedal triangle is the medial triangle. (Dan Reznik, July 7, 2020).

[1] Jakob Steiner , “Krümmungs Schwerpunkt ebener Curven” , Abhandlungen der Königlich Preussischen Akademie der Wissenschaften , 1838. (Dan Reznick, July 7, 2020)