X₂₉₁

2nd SHARYGIN POINT

X₂₉₁ = antipode of X₁ in hyperbola {A, B, C,X₁,X₂}

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

X₂₉₁ (2nd SHARYGIN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 39}, {2, 38}

• Related Centers: X₂₃₈ (-LINE CONJUGATE OF X₃₇), X₁ (INCENTER), X₃₉ (BROCARD MIDPOINT), X₂ (CENTROID), X₃₈ (CROSSPOINT OF X₁ AND X₇₅)

Trilinears: \(1/(a^{2} - bc) : 1/(b^{2} - ca) : 1/(c^{2} - ab)\)

Barycentrics: \(a/(a^{2} - bc) : b/(b^{2} - ca) : c/(c^{2} - ab)\)

Search Value (6, 9, 13): \(-0.192270503054\)

▸ Incident Central Lines14 lines

{1, 39}, {2, 38}, {6, 985}, {8, 330}, {10, 274}, {42, 81}, {43, 57}, {88, 660}, {105, 238}, {256, 894}, {337, 986}, {350, 726}, {659, 897}, {876, 891}

▸ Functional Relations11 relations

• X₂₉₁ = reflection of X_i in X_j for these (i,j): (1, 1015), (668, 10)

• X₂₉₁ = isogonal conjugate of X₂₃₈

• X₂₉₁ = isotomic conjugate of X₃₅₀

• X₂₉₁ = X₂-Ceva conjugate of X₃₆₉₀₆

• X₂₉₁ = X_i-cross conjugate of X_j for these (i,j): (239, 256), (518, 1)

• X₂₉₁ = X_i-Hirst inverse of X_j for these (i,j): (1, 292), (2, 335)

• X₂₉₁ = trilinear pole of PU(i) for these i: 6, 52, 53

• X₂₉₁ = antipode of X₁ in hyperbola {A, B, C,X₁,X₂}

• X₂₉₁ = X₁₉-isoconjugate of X₂₀₇₆₉

• X₂₉₁ = point of intersection, other than A, B, C, of 1st and 2nd bicentrics of the circumcircle

• X₂₉₁ = crossdifference of every pair of points on line X₆₅₉X₄₄₃₅ (the perspectrix of ABC and Gemini triangle 33)

Notes

See the description at X₁₂₈₁. The lines AD', BE', CF' defined there concur in X₂₉₁.