X₃₆₁₅

2nd HUTSON-FEUERBACH POINT

X₃₆₁₅ = cevapoint of X₁ and X₅

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

X₃₆₁₅ (2nd HUTSON-FEUERBACH POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {5, 49}, {21, 36}

• Related Centers: X₁ (INCENTER), X₅ (NINE-POINT CENTER), X₄₉ (CENTER OF SINE-TRIPLE-ANGLE CIRCLE), X₂₁ (SCHIFFLER POINT), X₃₆ (INVERSE-IN-CIRCUMCIRCLE OF INCENTER)

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

Barycentrics: \((\sin(A))/(\cos(B - A) + \cos(C - A)) : (\sin(B))/((\cos(C - B) + \cos(A - B)) : (\sin(C))/(\cos(A - C) + \cos(B - C))\)

Search Value (6, 9, 13): \(0.242505726184\)

▸ Alternate Coordinates1 form

• Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc(b + c - a)/[(b + c)(b^{2} + c^{2} - a^{2} + bc)]\) (Peter Moses, September 5, 2011)

▸ Incident Central Lines14 lines

{1, 564}, {2, 582}, {5, 49}, {11, 60}, {12, 59}, {21, 36}, {29, 1870}, {58, 3582}, {86, 1443}, {476, 953}, {501, 3583}, {655, 2595}, {946, 1325}, {2287, 2323}

▸ Functional Relations5 relations

• X₃₆₁₅ = isogonal conjugate of X₂₅₉₄

• X₃₆₁₅ = cevapoint of X₁ and X₅

• X₃₆₁₅ = intersection of tangents at X₁ and X₅ to hyperbola passing through X₁, X₅ and the excenters

• X₃₆₁₅ = crosspoint of X₁ and X₅ wrt the excentral triangle

• X₃₆₁₅ = trilinear pole of line X₆₅₄X₁₀₂₁

Notes

Let Eu_aEu_bEu_c be the Feuerbach triangle, L the line through Eu_a and X₅, and A″ = L∩BC; define B″ and C″ cyclically. Then the lines AA″, BB″, CC″ concur in X₃₆₁₅. (Randy Hutson, August 30, 2011.)

Let Eu_aEu_bEu_c be the Feuerbach triangle. Let A″ be the trilinear pole of line Eu_bEu_c, and define B″ and C″ cyclically. (The lines AA″, BB″, CC″ concur in X₁.) Let A* be the trilinear pole of line B″C″, and define B*and C* cyclically. The lines AA*, BB*, CC* concur in X₃₆₁₅. (Randy Hutson, July 11, 2019)