X₁₄₁

COMPLEMENT OF SYMMEDIAN POINT

X₁₄₁ is the complement of symmedian point.

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

X₁₄₁ (COMPLEMENT OF SYMMEDIAN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 6}

• Related Centers: X₂ (CENTROID), X₆ (SYMMEDIAN POINT)

Trilinears: \(bc(b^{2} + c^{2}) : :\)

Barycentrics: \(b^{2} + c^{2} : :\)

Search Value (6, 9, 13): \(3.447598941200\)

▸ Alternate Coordinates3 forms

• Trilinears: \(\csc^{2} A \sin(A + \omega ) : :\)

• Barycentrics: \(\cot A + \cot \omega : :\)

• Barycentrics: \(S_{A} + S_{\omega} : :\)

▸ Linear Combinations1 combination

• X₁₄₁ = \(3\) X₂ + X₆₉ = \(3\) X₂ - X₆

X₁₄₁ lies on the bicevian conic of X₂ and X₁₁₀.

▸ Incident Central Lines31 lines

{2, 6}, {3, 66}, {5, 211}, {10, 142}, {37, 742}, {39, 732}, {45, 344}, {53, 264}, {67, 110}, {75, 334}, {76, 698}, {95, 287}, {99, 755}, {113, 127}, {116, 121}, {125, 126}, {140, 182}, {239, 319}, {241, 307}, {308, 670}, {311, 338}, {317, 458}, {320, 894}, {384, 1031}, {441, 577}, {498, 611}, {499, 613}, {523, 882}, {542, 549}, {575, 629}, {997, 1060}

▸ Functional Relations33 relations

• X₁₄₁ = midpoint of X_i and X_j for these (i,j): (1, 3416), (6, 69), (8, 3242), (66, 159), (67, 110), (69, 3313), (1843, 3313) (2930, 3448)

• X₁₄₁ = reflection of X_i in X_j for these (i,j): (182, 140), (597, 2), (1353, 575), (1386, 1125)

• X₁₄₁ = isogonal conjugate of X₂₅₁

• X₁₄₁ = isotomic conjugate of X₈₃

• X₁₄₁ = inverse-in-nine-point-circle of X₆₂₅

• X₁₄₁ = complement of X₆

• X₁₄₁ = complementary conjugate of X₂

• X₁₄₁ = X_i-Ceva conjugate of X_j for these (i,j): (2, 39), (67, 524), (110, 525)

• X₁₄₁ = X₃₉-cross conjugate of X₄₂₇

• X₁₄₁ = crosspoint of X₂ and X₇₆

• X₁₄₁ = crosssum of X₆ and X₃₂

• X₁₄₁ = X₃₉-Hirst inverse of X₇₃₂

• X₁₄₁ = X₆₄₅-beth conjugate of X₁₄₁

• X₁₄₁ = X₆-of-medial triangle

• X₁₄₁ = anticomplement of X₃₅₈₉

• X₁₄₁ = centroid of ABCX₆₉

• X₁₄₁ = Kosnita(X₆₉,X₂) point

• X₁₄₁ = perspector of circumconic centered at X₃₉

• X₁₄₁ = center of circumconic that is locus of trilinear poles of lines passing through X₃₉

• X₁₄₁ = bicentric sum of PU(11)

• X₁₄₁ = midpoint of PU(11)

• X₁₄₁ = polar conjugate of X₃₂₀₈₅

• X₁₄₁ = X₆-of-X₂-Brocard-triangle

• X₁₄₁ = X₁₁₅-of-1st-Brocard-triangle

• X₁₄₁ = crosspoint of X₂ and X₂₈₉₆ wrt excentral triangle

• X₁₄₁ = crosspoint of X₂ and X₂₈₉₆ wrt anticomplementary triangle

• X₁₄₁ = crosspoint of X₆ and X₂₉₁₆ wrt excentral triangle

• X₁₄₁ = crosspoint of X₆ and X₂₉₁₆ wrt tangential triangle

• X₁₄₁ = {X₂,X₆}-harmonic conjugate of X₃₅₈₉

• X₁₄₁ = {X₃₉₅,X₃₉₆}-harmonic conjugate of X₅₃₀₆

• X₁₄₁ = trilinear cube root of X₁₄₁₂₅

• X₁₄₁ = perspector of 2nd Brocard triangle and cross-triangle of ABC and 2nd Brocard triangle

• X₁₄₁ = intersection, other than X₃, of the orthosymmedial circles of the 1st and 2nd Ehrmann inscribed triangles

Notes

Let P be a point on the circumcircle, and let L be the line tangent to the circumcircle at P. Let P' be the trilinear pole of L, and let P″ be the isotomic conjugate of P'. As P traces the circumcircle, P″ traces an ellipse inscribed in ABC with center at X₁₄₁. (Randy Hutson, December 26, 2015)

Let A'B'C' be the 2nd Brocard triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁₄₁. (Randy Hutson, December 26, 2015)

For an artistic design generated by X₁₄₁, see X₂₄₄.

X₁₄₁ is the {X₂,X₆₉}-harmonic conjugate of X₆.