X₅₆₁

ISOGONAL CONJUGATE OF 4th POWER POINT

X₅₆₁ is the isogonal conjugate of 4th power point.

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

X₅₆₁ (ISOGONAL CONJUGATE OF 4th POWER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {38, 75}, {92, 304}

• Related Centers: X₅₆₀ (4th POWER POINT), X₃₈ (CROSSPOINT OF X₁ AND X₇₅), X₇₅ (ISOTOMIC CONJUGATE OF INCENTER), X₉₂ (CEVAPOINT OF INCENTER AND CLAWSON POINT), X₃₀₄ (ISOTOMIC CONJUGATE OF X₁₉)

Trilinears: \(a ^{- 4} : b ^{- 4} : c ^{- 4}\)

Barycentrics: \(a ^{-3} : b ^{-3} : c^{-3}\)

Search Value (6, 9, 13): \(5.656128980924\)

▸ Alternate Coordinates2 forms

• Trilinears: \([\sec^{2}(A/2) + \csc^{2}(A/2)]^{2} : :\)

• Trilinears: \(d(a,b,c) : :\), where d(a,b,c) = \((distance from A to Lemoine axis)^{2} : :\)

▸ Incident Central Lines11 lines

{1, 718}, {2, 716}, {6, 720}, {31, 722}, {32, 724}, {38, 75}, {63, 799}, {76, 321}, {92, 304}, {313, 696}, {305, 1441}

▸ Functional Relations19 relations

• X₅₆₁ = isogonal conjugate of X₅₆₀

• X₅₆₁ = isotomic conjugate of X₃₁

• X₅₆₁ = complement of X₁₇₄₈₆

• X₅₆₁ = anticomplement of X₁₆₅₈₄

• X₅₆₁ = anticomplementary conjugate of anticomplement of X₃₈₈₁₀

• X₅₆₁ = antigonal conjugate of X₃₇₈₄₄

• X₅₆₁ = cevapoint of X₇₅ and X₃₀₄

• X₅₆₁ = X₃₁₃-cross conjugate of X₇₆

• X₅₆₁ = trilinear square of X₇₅

• X₅₆₁ = trilinear product of PU(11)

• X₅₆₁ = pole wrt polar circle of trilinear polar of X₁₉₇₃

• X₅₆₁ = X₄₈-isoconjugate (polar conjugate) of X₁₉₇₃

• X₅₆₁ = trilinear product of vertices of Gemini triangle 21

• X₅₆₁ = trilinear product of vertices of Gemini triangle 22

• X₅₆₁ = perspector of ABC and cross-triangle of Gemini triangles 31 and 32

• X₅₆₁ = perspector of ABC and cross-triangle of ABC and Gemini triangle 31

• X₅₆₁ = perspector of ABC and cross-triangle of ABC and Gemini triangle 32

• X₅₆₁ = barycentric product of vertices of Gemini triangle 31

• X₅₆₁ = barycentric product of vertices of Gemini triangle 32

Notes

X₅₆₁ is the vertex conjugate of the foci of the inellipse that is the trilinear square of the de Longchamps line (with center X₂₁₂₃₅ and perspector X₁₉₂₈). (Randy Hutson, October 15, 2018)

Let A'B'C' be the obverse triangle of X₁. Let A″B″C″ be the N-obverse triangle of X₁. Let A* be the trilinear product A'*A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₅₆₁. (Randy Hutson, October 15, 2018)