X₁₁₀₇

CROSSPOINT OF X₁ AND X₂₇₄

X₁₁₀₇ is the crosspoint of X₁ and X₂₇₄.

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

X₁₁₀₇ (CROSSPOINT OF X₁ AND X₂₇₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 6}, {10, 39}

• Related Centers: X₁ (INCENTER), X₂₇₄ (ISOGONAL CONJUGATE OF X₂₁₃), X₆ (SYMMEDIAN POINT), X₁₀ (SPIEKER CENTER), X₃₉ (BROCARD MIDPOINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(b^{2} + c^{2}) + bc(b + c)\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a,b,c)\)

Search Value (6, 9, 13): \(2.276737186100\)

▸ Linear Combinations2 combinations

• X₁₁₀₇ = \((\tan \omega \sin 2\omega )R\) X₁₀ + \(r\) X₃₉

• X₁₁₀₇ = X₁ + X₈ + \((\cot \omega \csc 2\omega )(2r/R)\) X₃₉

▸ Incident Central Lines7 lines

{1, 6}, {2, 330}, {10, 39}, {32, 993}, {75, 194}, {210, 869}, {239, 257}

▸ Functional Relations10 relations

• X₁₁₀₇ = midpoint of the bicentric pair \(y : z : x\) and \(z : x : y\), where \(x : y : z\) = X₂₁₃

• X₁₁₀₇ = insimilicenter of the Spieker and (1/2)-Moses circle. [The (1/2)-Moses circle is described at X₁₅₇₅.]

• X₁₁₀₇ = isogonal conjugate of X₁₂₅₈

• X₁₁₀₇ = isotomic conjugate of X₁₂₂₁

• X₁₁₀₇ = complement of X₁₉₀₉

• X₁₁₀₇ = crosspoint of X_i and X_j for these (i,j): (1, 274), (2, 256), (81, 87)

• X₁₁₀₇ = crosssum of X_i and X_j for these (i,j): (1, 213), (6, 171), (37, 43)

• X₁₁₀₇ = polar conjugate of isogonal conjugate of X₂₂₃₈₉

• X₁₁₀₇ = {X₁,X₉}-harmonic conjugate of X₂₁₇₆

• X₁₁₀₇ = {X₁₀,X₃₉}-harmonic conjugate of X₁₅₇₅