X₁₈₄₃

ZOSMA TRANSFORM OF X₇₅

X₁₈₄₃ = orthic isotomic conjugate of X₁₈₅

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

X₁₈₄₃ (ZOSMA TRANSFORM OF X₇₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 25}, {4, 69}

• Related Centers: X₁₇₉₉ (INVERSE MIMOSA TRANSFORM OF X₇₅), X₆ (SYMMEDIAN POINT), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₄ (ORTHOCENTER), X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE)

Trilinears: \((y + z) \sec A : (z + x) \sec B : (x + y) \sec C\), where \(x : y : z\) = X₇₅

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

Search Value (6, 9, 13): \(-1.623940242009\)

▸ Alternate Coordinates1 form

• Trilinears: \(\tan A \sin(A + \omega ) : :\)

▸ Incident Central Lines22 lines

{4, 69}, {6, 25}, {24, 182}, {34, 1469}, {112, 755}, {113, 1596}, {125, 1205}, {141, 427}, {143, 1353}, {155, 1351}, {157, 571}, {160, 570}, {181, 1395}, {185, 1503}, {216, 237}, {263, 393}, {373, 468}, {428, 524}, {518, 1829}, {674, 1824}, {1350, 1593}, {1827, 1828}

▸ Functional Relations15 relations

• X₁₈₄₃ = reflection of X_i in X_j for these (i,j): (1205, 125), (1289, 647)

• X₁₈₄₃ = isogonal conjugate of X₁₇₉₉

• X₁₈₄₃ = X_i-Ceva conjugate for these (i,j): (4, 427), (427, 39), (1289, 647)

• X₁₈₄₃ = crosspoint of X_i and X_j for these (i,j): (4, 25), (6, 66)

• X₁₈₄₃ = crosssum of X_i and X_j for these (i,j): (2, 22), (3, 69)

• X₁₈₄₃ = orthic isotomic conjugate of X₁₈₅

• X₁₈₄₃ = X₇-of-orthic-triangle if ABC is acute

• X₁₈₄₃ = orthic-isogonal conjugate of X₄₂₇

• X₁₈₄₃ = anticomplement of X₆ wrt orthic triangle

• X₁₈₄₃ = pole wrt polar circle of trilinear polar of X₃₀₈ (line X₃₁₆X₅₁₂)

• X₁₈₄₃ = X₄₈-isoconjugate (polar conjugate) of X₃₀₈

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

• X₁₈₄₃ = excentral-to-ABC functional image of X₇

• X₁₈₄₃ = Ehrmann-vertex-to-orthic similarity image of X₃₈₁₈

• X₁₈₄₃ = perspector of [reflection of symmedial triangle in X₆] and tangential triangle, wrt symmedial triangle, of circumconic of symmedial triangle centered at X₆ (bicevian conic of X₆ and X₂₅)

Notes

Let E = Euler line of ABC. Let g = isogonal conjugate, and t = isotomic conjugate. Then X₁₈₄₃ = g(t(E))∩t(g(E)). (Randy Hutson, December 2, 2017)