X₃₁₆

DROUSSENT PIVOT

X₃₁₆ = anticomplement of X₁₈₇

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

X₃₁₆ (DROUSSENT PIVOT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {621, 622}

• Related Centers: X₆₂₁ (ANTICOMPLEMENT OF X₁₅), X₆₂₂ (ANTICOMPLEMENT OF X₁₆)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc(b^{4} + c^{4} - a^{4} - b^{2}c^{2})\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = b^{4} + c^{4} - a^{4} - b^{2}c^{2}\)

Tripolars: \(a^{2} \sqrt{2(b^{2} + c^{2}) - a^{2}} : :\)

Search Value (6, 9, 13): \(11.569873852650\)

▸ Incident Central Lines17 lines

{2, 187}, {4, 69}, {15, 303}, {16, 302}, {30, 99}, {115, 385}, {148, 538}, {183, 381}, {249, 297}, {265, 290}, {298, 530}, {299, 531}, {376, 1007}, {384, 626}, {512, 850}, {524, 671}, {691, 858}

▸ Functional Relations16 relations

• X₃₁₆ = midpoint of X₆₂₁ and X₆₂₂

• X₃₁₆ = reflection of X_i in X_j for these (i,j): (15, 624), (16, 623), (99, 325), (385, 115), (691, 858)

• X₃₁₆ = isogonal conjugate of X₃₄₅₅

• X₃₁₆ = isotomic conjugate of X₆₇

• X₃₁₆ = anticomplement of X₁₈₇

• X₃₁₆ = crosssum of X₃₉ and X₁₈₇

• X₃₁₆ = reflection of X₉₉ in the polar of X₇₆

• X₃₁₆ = antigonal conjugate of X₂₃

• X₃₁₆ = reflection of X₉₉ in the de Longchamps line

• X₃₁₆ = inverse-in-polar-circle of X₁₈₄₃

• X₃₁₆ = trilinear pole of line X₂₄₉₂X₇₆₆₄

• X₃₁₆ = pole wrt polar circle of trilinear polar of X₈₇₉₁ (line X₅₁₂X₁₈₄₃)

• X₃₁₆ = polar conjugate of X₈₇₉₁

• X₃₁₆ = inverse-in-circumcircle of X₂₁₃₉₅

• X₃₁₆ = intersection of Lemoine axes of 1st and 2nd Ehrmann circumscribing triangles

• X₃₁₆ = intersection of Lemoine axes of anticevian triangles of PU(4)

Notes

Let K_aK_bK_c be the antipedal triangle of X₄ (the anticomplementary triangle). The circumcircles of AK_aX₄, BK_bX₄, CK_cX₄ concur, other than X₄, in X₃₁₆. (Randy Hutson, June 27, 2018)

See Lucien Droussent, “Cubiques circulaires anallagmatiques par points réciproques ou isogonaux,” Mathesis 62 (1953) 204-215.