X₁₀₁₅

EXSIMILICENTER OF MOSES CIRCLE AND INCIRCLE

The circle having center X₃₉ and radius 2R \(\sin^{2}\)ω, where R denotes the circumradius of triangle ABC, is here introduced as the Moses circle. It is tangent to the nine-point circle at X₁₁₅, and its internal and external centers of similitude with the incircle are X₁₅₀₀ and X₁₀₁₅, respectively. (based on notes from Peter J. C. Moses, 5/29/03)

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

X₁₀₁₅ (EXSIMILICENTER OF MOSES CIRCLE AND INCIRCLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 39}, {32, 56}

• Related Centers: X₁₀₁₆ (-CROSS CONJUGATE OF X₉₉), X₁ (INCENTER), X₃₉ (BROCARD MIDPOINT), X₃₂ (3rd POWER POINT), X₅₆ (EXSIMILICENTER)

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

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

Search Value (6, 9, 13): \(0.749019003202\)

▸ Linear Combinations1 combination

• X₁₀₁₅ = \((\tan \omega \sin 2\omega )R\) X₁ - \(r\) X₃₉

X₁₀₁₅ lies on the Steiner inellipse, the Brocard inellipse.

▸ Incident Central Lines15 lines

{1, 39}, {2, 668}, {6, 101}, {11, 115}, {32, 56}, {36, 187}, {37, 537}, {55, 574}, {76, 330}, {214, 1100}, {216, 1060}, {244, 665}, {350, 538}, {812, 1086}, {1960, 3122}

▸ Functional Relations19 relations

• X₁₀₁₅ = midpoint of X_i and X_j for these (i,j): (1, 291), (2, 3227)

• X₁₀₁₅ = isogonal conjugate of X₁₀₁₆

• X₁₀₁₅ = isotomic conjugate of X₃₁₆₂₅

• X₁₀₁₅ = complement of X₆₆₈

• X₁₀₁₅ = crosspoint of X₂ and X₅₁₃

• X₁₀₁₅ = crosssum of X_i and X_j for these (i,j): (1, 1018), (2, 190), (6, 100), (8, 644), (101, 595), (345, 1332)

• X₁₀₁₅ = crosssum of circumcircle-intercepts of line X₁X₆

• X₁₀₁₅ = crossdifference of every pair of points on line X₁₀₀X₁₉₀

• X₁₀₁₅ = PU(25)-harmonic conjugate of X₁₉₆₀

• X₁₀₁₅ = polar conjugate of isogonal conjugate of X₂₂₀₉₆

• X₁₀₁₅ = bicentric difference of PU(25)

• X₁₀₁₅ = bicentric sum of PU(27)

• X₁₀₁₅ = midpoint of PU(27)

• X₁₀₁₅ = center of circumconic that is locus of trilinear poles of lines through X₅₁₃

• X₁₀₁₅ = perspector of circumparabola centered at X₅₁₃

• X₁₀₁₅ = X₂-Ceva conjugate of X₅₁₃

• X₁₀₁₅ = projection from Steiner circumellipse to Steiner inellipse of X₃₂₂₇

• X₁₀₁₅ = barycentric square of X₅₁₃

• X₁₀₁₅ = center of {ABC, Gemini 7}-circumconic

Notes

X₁₀₁₅ is the center of the hyperbola that passes through the points A, B, C, X₁, X₂, and X₂₈; also, X₁₀₁₅ lies on the ellipse described at X₁₁₂₅. (Randy Hutson, Hyacinthos #20179, 8/13/2011)