X₃₃₁₁

INTERSECTION OF LINES X₃X_6 AND X₄X_1131

X₃₃₁₁ = inverse-in-Brocard-circle of X₃₃₁₂

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

X₃₃₁₁ (INTERSECTION OF LINES X₃X_6 AND X₄X_1131) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {25, 588}

• Related Centers: X₃₃₁₆ (ISOGONAL CONJUGATE OF X₃₃₁₁), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₅₈₈ (1st KENMOTU-VAN LAMOEN POINT)

Trilinears: \(\cos A + 2 \sin A : \cos B + 2 \sin B : \cos C + 2 \sin C\)

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

Search Value (6, 9, 13): \(1.814941052584\)

▸ Alternate Coordinates4 forms

• Trilinears: \(\cos(A - t) : \cos(B - t) : \cos(C - t)\), where t = arctan(2)

• Trilinears: \(a + R \cos A : :\)

• Trilinears: \(a(S_{A} + 2S) : :\)

• Trilinears: \(a(b^{2} + c^{2} - a^{2} + 4S) : :\)

▸ Linear Combinations1 combination

• X₃₃₁₁ = \(La'/R_{a}' + Lb'/R_{b}' + Lc'/R_{c}'\), where L_a', L_b', L_c' are the centers of the secondary Lucas circles, and \(R_{a}'\), \(R_{b}'\), \(R_{c}'\) are their radii

▸ Incident Central Lines12 lines

{3, 6}, {4, 1131}, {5, 1588}, {25, 588}, {30, 1587}, {381, 485}, {486, 590}, {488, 1992}, {517, 1702}, {591, 641}, {640, 1991}, {999, 1124}

▸ Functional Relations7 relations

• X₃₃₁₁ = isogonal conjugate of X₃₃₁₆

• X₃₃₁₁ = inverse-in-Brocard-circle of X₃₃₁₂

• X₃₃₁₁ = {X₆,X₃₇₁}-harmonic conjugate of X₃

• X₃₃₁₁ = {X₆,X₁₁₅₁}-harmonic conjugate of X₃₇₂

• X₃₃₁₁ = {X₆₁,X₆₂}-harmonic conjugate of X₃₅₉₂

• X₃₃₁₁ = {X₃₇₁,X₃₇₂}-harmonic conjugate of X₁₁₅₁

• X₃₃₁₁ = {X₃₇₂,X₁₁₅₁}-harmonic conjugate of X₃

Notes

X₃₃₁₁ is the perspector of each of the following pairs of triangles:

• Lucas central triangle and the symmedial triangle (the cevian triangle of X₆)

• Lucas tangents triangle and the Lucas(-1:1) central triangle

• Lucas(2:3) central triangle and the circumsymmedial triangle.

Moreover, X₃₃₁₁, is the radical center of the Lucas(4:1) circles. See X₃₇₁ and X₃₃₁₂. (Randy Hutson, 9/23/2011)

X₃₃₁₁ is the perspector of each of the pair of the following four triangles:

• symmedial triangle. Lucas central triangle, Lucas(-1) secondary central triangle, 1st Lucas(-1) secondary tangents triangle. (Randy Hutson, September 14,. 2016)