X₂₁₆

X₅-CEVA CONJUGATE OF X₅₁

X₂₁₆ = X₂₁₇-cross conjugate of X₅₁

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

X₂₁₆ (X₅-CEVA CONJUGATE OF X₅₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {5, 53}

• Related Centers: X₅ (NINE-POINT CENTER), X₅₁ (CENTROID OF ORTHIC TRIANGLE), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₅₃ (SYMMEDIAN POINT OF ORTHIC TRIANGLE)

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

Barycentrics: \(\csc 2B + \csc 2C : :\)

Search Value (6, 9, 13): \(-0.485600261779\)

▸ Alternate Coordinates9 forms

• Trilinears: \(\cos A (\tan A + \tan B + \tan C) + \sin A : :\)

• Trilinears: \(\cos A + \sin A (\cot A \cot B \cot C) : :\)

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

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

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

• Trilinears: \(\sec B \sec C + \csc B \csc C : :\)

• Trilinears: \(\cos(A + T) : :\), T as at X₃₈₉

• Barycentrics: \(a^{2}(b^{2} + c^{2} - a^{2})[a^{2}(b^{2} + c^{2}) - (b^{2} - c^{2})^{2}] : :\)

• Barycentrics: \((\sin A)(\sin 2A)\cos(B - C) : :\)

X₂₁₆ lies on hyperbola {X₂,X₆,X₂₁₆,X₂₃₃,X₁₂₄₉,X₁₅₆₀,X₃₁₆₂}}, which is a circumconic of the medial triangle, as well as the locus of the perspector of circumconics centered at a point on the Euler line. Also, this hyperbola is tangent to Euler line at X₂. (Randy Hutson, July 20, 2016)

▸ Incident Central Lines14 lines

{2, 232}, {3, 6}, {5, 53}, {51, 418}, {95, 648}, {97, 288}, {115, 131}, {157, 206}, {373, 852}, {395, 465}, {396, 466}, {631, 1075}, {1015, 1060}, {2493, 3054}

▸ Functional Relations23 relations

• X₂₁₆ = intersection of isogonal conjugate of polar conjugate of Euler line (i.e., line X₃X₆) and the polar conjugate of isogonal conjugate of Euler line (i.e., line X₂X₂₁₆) (Randy Hutson, July 20, 2016)

• X₂₁₆ = isogonal conjugate of X₂₇₅

• X₂₁₆ = isotomic conjugate of X₂₇₆

• X₂₁₆ = inverse-in-Brocard-circle of X₅₇₇

• X₂₁₆ = complement of X₂₆₄

• X₂₁₆ = complementary conjugate of X₂₁₂₄₃

• X₂₁₆ = X_i-Ceva conjugate of X_j for these (i,j): (2, 5), (3, 418), (5, 51), (324, 52)

• X₂₁₆ = cevapoint of X₂₁₇ and X₄₁₈

• X₂₁₆ = X₂₁₇-cross conjugate of X₅₁

• X₂₁₆ = crosspoint of X_i and X_j for these (i,j): (2, 3), (5, 343)

• X₂₁₆ = crosssum of X₄ and X₆

• X₂₁₆ = crossdifference of every pair of points on line X₁₈₆X₅₂₃

• X₂₁₆ = inverse-in-Brocard-circle of X₅₇₇

• X₂₁₆ = center of circumconic that is locus of trilinear poles of lines passing through X₅

• X₂₁₆ = intersection of trilinear polars of any 2 points on the Johnson circumconic

• X₂₁₆ = perspector of cevian triangle of X₃ and tangential triangle, wrt cevian triangle of X₃, of circumconic of cevian triangle of X₃ centered at X₃

• X₂₁₆ = pole wrt polar circle of trilinear polar of X₈₇₉₅

• X₂₁₆ = X₄₈-isoconjugate (polar conjugate) of X₈₇₉₅

• X₂₁₆ = X₉₂-isoconjugate of X₅₄

• X₂₁₆ = X₁₅₇₇-isoconjugate of X₉₃₃

• X₂₁₆ = X₅₇₃-of-orthic-triangle if ABC is acute

• X₂₁₆ = perspector of ABC and unary cofactor triangle of circumorthic triangle

• X₂₁₆ = {X₆₁,X₆₂}-harmonic conjugate of X₃₇₅₀₅

Notes

X₂₁₆ is the perspector of triangle ABCand the tangential triangle of the Johnson circumconic. (Randy Hutson, 9/23/2011)

Let E_a be the ellipse with B and C as foci and passing through X₅, and define E_b, E_c cyclically. Let L_a be the line tangent to E_a at X₅, and define L_b, L_c cyclically. Let A' be the trilinear pole of line L_a, and define B', C' cyclically. A', B', C' lie on the circumconic centered at X₂₁₆. (Randy Hutson, July 20, 2016)