X₄₇

X₁₁₀-BETH CONJUGATE OF X₃₄

X₄₇ = perspector of ABC and extraversion triangle of X₄₇ (which is also the anticevian triangle of X₄₇)

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

X₄₇ (X₁₁₀-BETH CONJUGATE OF X₃₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 21}, {34, 46}

• Related Centers: X₁₁₀ (FOCUS OF KIEPERT PARABOLA), X₃₄ (-BETH CONJUGATE OF X₄), X₁ (INCENTER), X₂₁ (SCHIFFLER POINT), X₄₆ (-CEVA CONJUGATE OF X₁)

Trilinears: \(\cos 2A : \cos 2B : \cos 2C = f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a^{2}(a^{4} + b^{4} + c^{4} - 2a^{2}b^{2} - 2a^{2}c^{2})\)

Barycentrics: \(a \cos 2A : b \cos 2B : c \cos 2C\)

Search Value (6, 9, 13): \(-56.512488734291\)

▸ Alternate Coordinates5 forms

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

• Trilinears: \(\tan A \cot 2A : :\)

• Trilinears: \(\cos^{2} A - \sin^{2} A : :\)

• Trilinears: \(1 - 2 \sin^{2} A : :\)

• Trilinears: \(1 - 2 \cos^{2} A : :\)

▸ Linear Combinations1 combination

• X₄₇ = \((r^{2} - R^{2} + s^{2})\) X₁ - \(6rR\) X₂ - \(4r^{2}\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines10 lines

{1, 21}, {19, 921}, {33, 90}, {34, 46}, {35, 212}, {36, 602}, {91, 92}, {158, 162}, {171, 498}, {238, 499}

▸ Functional Relations13 relations

• X₄₇ = isogonal conjugate of X₉₁

• X₄₇ = isotomic conjugate of X₂₀₅₇₁

• X₄₇ = anticomplement of X₃₄₈₂₅

• X₄₇ = trilinear product X₃₇₁*X₃₇₂

• X₄₇ = X₉₂-isoconjugate of X₁₈₂₀

• X₄₇ = perspector of ABC and extraversion triangle of X₄₇ (which is also the anticevian triangle of X₄₇)

• X₄₇ = eigencenter of cevian triangle of X₉₂

• X₄₇ = eigencenter of anticevian triangle of X₄₈

• X₄₇ = X₉₂-Ceva conjugate of X₄₈

• X₄₇ = crosssum of X_i and X_j for these (i,j): (656, 1109)

• X₄₇ = X₂₇₅-aleph conjugate of X₉₂

• X₄₇ = X_i-beth conjugate of X_j for these (i,j): (110, 34), (643, 47)

• X₄₇ = trilinear product of X₃₇₁ and X₃₇₂

Notes

Let O_AO_BO_C be the Kosnita triangle. Let A″ be the trilinear product O_B*O_C, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₄₇. (Randy Hutson, March 21, 2019)

Let A'B'C' and A″B″C″ be the Lucas and Lucas(-1) tangents triangles. Let A* be the trilinear product A'*A″, and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X₄₇. (Randy Hutson, March 21, 2019)

X₄₇ is the {X₉₁,X₉₂}-harmonic conjugate of X₅₆₄.