X₅₇₈

ORTHOHARMONIC OF X₂₁₆

X₅₇₈ = inverse-in-Brocard-circle of X₃₈₉

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

X₅₇₈ (ORTHOHARMONIC OF X₂₁₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {4, 54}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₄ (ORTHOCENTER), X₅₄ (KOSNITA POINT)

Trilinears: \(\sin A + \cot D/2 \cos A\), where \(\cot(D/2) = (a^{2}b^{2}c^{2} \cos A \cos B \cos C)/[8 (\operatorname{area})^{3}]\)

Barycentrics: \(a^{2}[a^{8} - 3a^{6}(b^{2} + c^{2}) + a^{4}(3b^{4} + 4b^{2}c^{2} + 3c^{4}) - a^{2}(b^{2} - c^{2})^{2}(b^{2} + c^{2}) - 2b^{2}c^{2}(b^{2} - c^{2})^{2}] : :\)

Search Value (6, 9, 13): \(-5.761794368781\)

▸ Alternate Coordinates2 forms

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

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

▸ Incident Central Lines8 lines

{2, 1092}, {3, 6}, {4, 54}, {24, 51}, {49, 381}, {156, 546}, {185, 378}, {436, 1093}

▸ Functional Relations2 relations

• X₅₇₈ = inverse-in-Brocard-circle of X₃₈₉

• X₅₇₈ = X₁₂₅₁₄-of-orthic-triangle if ABC is acute

Notes

Joe Goggins notes (10/1/2008), in connection with the note at X₃₈₉, that trilinears for X₅₇₈ are \(\sin(A-T) : \sin(B-T) : \sin(C-T)\), where tan(T) = - cot A cot B cot C.

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the Euler triangle at X₅₇₈. (Randy Hutson, March 14, 2018)