X₃₀₅₃

INTERSECTION X₃X_6∩X₄X_230

X₃₀₅₃ = inverse-in-circumcircle of X₁₆₉₂

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

X₃₀₅₃ (INTERSECTION X₃X_6∩X₄X_230) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Central Lines: {3, 1692}

• Related Centers: X₃ (CIRCUMCENTER), X₁₆₉₂ (RADICAL TRACE OF CIRCUMCIRCLE AND 2nd LEMOINE CIRCLE)

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

Search Value (6, 9, 13): \(-0.631285612775\)

▸ Alternate Coordinates5 forms

• Trilinears: \(a(a^{2} - S_{A}) : :\)

• Trilinears: \(\sin A - 2 \cos A \tan \omega : :\)

• Trilinears: \(2 \cos A - \sin A \cot \omega : :\)

• Trilinears: \(a - 4R \cos A \tan \omega : :\)

• Trilinears: \(a (\cot A - \cot B \cot C) : :\)

▸ Incident Central Lines19 lines

{3, 6}, {4, 230}, {22, 1184}, {24, 112}, {25, 1611}, {35, 609}, {56, 1914}, {64, 248}, {76, 1003}, {115, 382}, {140, 2548}, {154, 237}, {218, 2251}, {550, 2549}, {988, 1100}, {999, 2241}, {1385, 1572}, {1498, 1971}, {2176, 2223}

▸ Functional Relations15 relations

• X₃₀₅₃ = isogonal conjugate of X₂₉₉₆

• X₃₀₅₃ = inverse-in-circumcircle of X₁₆₉₂

• X₃₀₅₃ = X₂₅-Ceva conjugate of X₆

• X₃₀₅₃ = crosspoint of X_i and X_j for these (i,j): (112, 249), (193, 459)

• X₃₀₅₃ = crosssum of X₁₁₅ and X₅₂₅

• X₃₀₅₃ = radical center of Lucas(-cot ω) circles

• X₃₀₅₃ = vertex, other than X₆, of the hyperbola {{X₆,PU(1),PU(2)}}

• X₃₀₅₃ = intersection of diagonals of trapezoid PU(2)PU(39)

• X₃₀₅₃ = insimilicenter of circles with diameters X₃₇₁X₃₇₂ and X₁₁₅₁X₁₁₅₂

• X₃₀₅₃ = trilinear pole wrt tangential triangle of orthic axis

• X₃₀₅₃ = crossdifference of every pair of points on line X₅₂₃X₄₈₈₅ (complement of orthic axis)

• X₃₀₅₃ = {X₁₆₈₇,X₁₆₈₈}-harmonic conjugate of X₅₁₇₁

• X₃₀₅₃ = Schoutte-circle-inverse of X₃₃₈₇₈

• X₃₀₅₃ = insimilicenter of circle centered at X₁₁₅₁ through X₃₇₂ and circle centered at X₁₁₅₂ through X₃₇₁; the exsimilicenter is X₁₃₅₀

• X₃₀₅₃ = insimilicenter of circle centered at X₃₇₁ through X₁₁₅₂ and circle centered at X₃₇₂ through X₁₁₅₁; the exsimilicenter is X₁₃₅₁

Notes

Let Γ₁ be the circumcircle, Γ₂ the 2nd Lemoine circle, and Γ₃ the circle {{X₃₇₁,X₃₇₂,PU(1),PU(39)}} (which has center X₃₂). The three circles intersect in two points, at which let L and L' be the lines tangent to Γ₁. Then X₃₀₃₅ = L∩L'. (Randy Hutson, January 29, 2015)