X₄₈₄

1st EVANS PERSPECTOR

X₄₈₄ is the perspector of the excentral triangle and the triangle Ref_aRef_bRef_c, where Ref_a is the reflection of vertex A in sideline BC and Ref_b, Ref_c are determined cyclically. (Lawrence Evans, 10/22/98)

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

X₄₈₄ (1st EVANS PERSPECTOR) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {36, 3245}

• Related Centers: X₃₆ (INVERSE-IN-CIRCUMCIRCLE OF INCENTER), X₃₂₄₅ (REFLECTION OF X)

Trilinears: \(1 + 2(\cos A - \cos B - \cos C) : 1 + 2(\cos B - \cos C - \cos A) : 1 + 2(\cos C - \cos A - \cos B)\)

Barycentrics: \(a[1 + 2(\cos A - \cos B - \cos C)] : b[1 + 2(\cos B - \cos C - \cos A)] : c[1 + 2(\cos C - \cos A - \cos B)]\)

Search Value (6, 9, 13): \(-6.840882082776\)

▸ Alternate Coordinates1 form

• Trilinears: \(a^{3} + a^{2} (b + c) - a (b^{2} + b c + c^{2}) - (b - c)^{2} (b + c) : :\)

X₄₈₄ lies on the Neuberg cubic.

▸ Incident Central Lines15 lines

{1, 3}, {4, 3483}, {10, 191}, {12, 79}, {13, 1277}, {14, 1276}, {30, 80}, {63, 535}, {74, 3465}, {100, 758}, {499, 962}, {759, 901}, {1046, 1048}, {1138, 3464}, {3466, 3484}

▸ Functional Relations16 relations

• X₄₈₄ = midpoint of X₃₆ and X₃₂₄₅

• X₄₈₄ = reflection of X_i in X_j for these (i,j): (1, 36), (36, 1155)

• X₄₈₄ = isogonal conjugate of X₃₀₆₅

• X₄₈₄ = inverse-in-circumcircle of X₃₅

• X₄₈₄ = inverse-in-Bevan-circle of X₁

• X₄₈₄ = Conway-circle-inverse of X₃₅₆₃₁

• X₄₈₄ = X₈₀-Ceva conjugate of X₁

• X₄₈₄ = crossdifference of every pair of points on line X₆₅₀X₁₁₀₀

• X₄₈₄ = {X₁X₄₆}-harmonic conjugate of X₃₃₃₆

• X₄₈₄ = inverse-in-incircle of X₅₀₄₅

• X₄₈₄ = perspector of ABC and the reflection of the incentral triangle in the antiorthic axis (the reflection of the cevian triangle of X₁ in the trilinear polar of X₁)

• X₄₈₄ = reflection of X₃₆ in the antiorthic axis

• X₄₈₄ = X₃₀-Ceva conjugate of X₃₄₆₅

• X₄₈₄ = X₁₈₆-of-excentral-triangle

• X₄₈₄ = {X₁,X₃}-harmonic conjugate of X₃₇₆₁₆

• X₄₈₄ = endo-homothetic center of Ehrmann vertex-triangle and dual of orthic triangle; the homothetic center of is X₃₁₅₃