X₂₁₂

X₉-CEVA CONJUGATE OF X₄₁

X₂₁₂ = X₄-isoconjugate of X₇

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

X₂₁₂ (X₉-CEVA CONJUGATE OF X₄₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 31}, {9, 33}

• Related Centers: X₉ (MITTENPUNKT), X₄₁ (-CEVA CONJUGATE OF X₃₁), X₆ (SYMMEDIAN POINT), X₃₁ (2nd POWER POINT), X₃₃ (PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES)

Trilinears: \((\cos A)(1 + \cos A) : (\cos B)(1 + \cos B) : (\cos C)(1 + \cos C)\)

Barycentrics: \((\sin 2A)(1 + \cos A) : (\sin 2B)(1 + \cos B) : (\sin 2C)(1 + \cos C)\)

Search Value (6, 9, 13): \(4.122746567678\)

▸ Alternate Coordinates2 forms

• Trilinears: \((\cos A)\cos^{2}(A/2) : (\cos B)\cos^{2}(B/2) : (\cos C)\cos^{2}(C/2)\)

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

▸ Incident Central Lines16 lines

{1, 201}, {3, 73}, {6, 31}, {9, 33}, {11, 748}, {34, 40}, {35, 47}, {48, 184}, {56, 939}, {63, 1040}, {78, 283}, {109, 165}, {154, 198}, {238, 497}, {312, 643}, {582, 942}

▸ Functional Relations9 relations

• X₂₁₂ = isogonal conjugate of X₂₇₃

• X₂₁₂ = crossdifference of every pair of points on line X₅₁₄X₃₀₆₄

• X₂₁₂ = X₄-isoconjugate of X₇

• X₂₁₂ = X₅₇-isoconjugate of X₉₂

• X₂₁₂ = X_i-Ceva conjugate of X_j for these (i,j): (3, 48), (9, 41), (283, 219)

• X₂₁₂ = X₂₂₈-cross conjugate of X₅₅

• X₂₁₂ = crosspoint of X_i and X_j for these (i,j): (3, 219), (9, 78)

• X₂₁₂ = crosssum of X_i and X_j for these (i,j): (4, 278), (34, 57)

• X₂₁₂ = X₂₁₂-beth conjugate of X₁₈₄

Notes

The trilinear polar of X₂₁₂ passes through X₁₉₄₆. (Randy Hutson, June 7, 2019)