X₁₆₅₂

4th EVANS PERSPECTOR

X₁₆₅₂ = X₅₅₄-aleph conjugate of X₁₆₅₂

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

X₁₆₅₂ (4th EVANS PERSPECTOR) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 7}, {1, 15}

• Related Centers: X₂ (CENTROID), X₇ (GERGONNE POINT), X₁ (INCENTER), X₁₅ (1st ISODYNAMIC POINT)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \sqrt{3/4} - \sin(A + \pi /3) + \sin(B + \pi /3) + \sin(C + \pi /3)\)

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(0.835552003148\)

▸ Alternate Coordinates3 forms

• Trilinears: \(\sqrt{3/4} - \cos(A - \pi /6) + \cos(B - \pi /6) + \cos(C - \pi /6)\)

• Trilinears: \(\sqrt{3} \tan(A/2) + \tan(B/2) \tan(C/2) : \sqrt{3} \tan(B/2) + \tan(C/2) \tan(A/2) : \sqrt{3} \tan(C/2) + \tan(A/2) \tan(B/2)\)

• Trilinears: \(\sqrt{3} \tan(A/2) + (b + c - a)/(a + b + c) : \sqrt{3} \tan(B/2) + (c + a - b)/(a + b + c) : \sqrt{3} \tan(C/2) + (a + b - c)/(a + b + c)\) (M. Iliev, 5/13/07)

▸ Incident Central Lines12 lines

{1, 15}, {2, 7}, {13, 3464}, {17, 3336}, {46, 1277}, {56, 5240}, {61, 3468}, {65, 5239}, {396, 554}, {559, 1100}, {3638, 5011}, {4848, 5245}

▸ Functional Relations3 relations

• X₁₆₅₂ = X_i-Ceva conjugate of X_j for these (i,j): (554, 1), (2160, 1653)

• X₁₆₅₂ = X₅₅₄-aleph conjugate of X₁₆₅₂

• X₁₆₅₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (1, 3179, 1276), (9, 57, 1653), (1400, 3218, 1653), (2285, 3306, 1653)

Notes

See also X₁₂₇₆ and X₁₂₇₇ for the 2nd and 3rd Evans perspectors.