X₃₀₉₅

INTERSECTION OF LINES X₃X_6 AND X₅X_76

X₃₀₉₅ = 2nd-Brocard-circle-inverse of X₁₈₂

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

X₃₀₉₅ (INTERSECTION OF LINES X₃X_6 AND X₅X_76) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {5, 76}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₅ (NINE-POINT CENTER), X₇₆ (3rd BROCARD POINT)

Trilinears: \(\cos(A + 2\omega ) : \cos(B + 2\omega ) : \cos(C + 2\omega )\)

Search Value (6, 9, 13): \(-3.653403601270\)

▸ Alternate Coordinates2 forms

• Trilinears: \(a(b^{6} + c^{6} - a^{2}b^{4} - a^{2}c^{4} - 3a^{2}b^{2}c^{2}) : :\)

• Trilinears: \(e^{2}\sin(A + \omega ) - \sin(A - \omega ) : :\)

▸ Incident Central Lines9 lines

{3, 6}, {4, 147}, {5, 76}, {237, 3060}, {355, 730}, {381, 538}, {726, 946}, {732, 1352}, {1993, 2001}

▸ Functional Relations15 relations

• X₃₀₉₅ = midpoint of X_i and X_j for these (i,j): (4, 194), (3557, 3558)

• X₃₀₉₅ = reflection of X_i in X_j for these (i,j): (3, 39), (76, 5)

• X₃₀₉₅ = isogonal conjugate of X₃₄₀₆

• X₃₀₉₅ = circumcircle-inverse of X₃₅₃₇₅

• X₃₀₉₅ = Brocard-circle-inverse of X₃₃₉₈

• X₃₀₉₅ = 2nd-Brocard-circle-inverse of X₁₈₂

• X₃₀₉₅ = reflection of X₃ in line PU(1)

• X₃₀₉₅ = harmonic center of 2nd Lemoine circle and the circle {{X₃₇₁,X₃₇₂,PU(1),PU(39)}}

• X₃₀₉₅ = center of circle {{X₄,X₁₉₄,X₃₅₅₇,X₃₅₅₈}}

• X₃₀₉₅ = X₄-of-X₆PU(1)

• X₃₀₉₅ = orthologic center of these triangles: Johnson to 1st Neuberg

• X₃₀₉₅ = X₇₆-of-Johnson-triangle

• X₃₀₉₅ = inverse-in-circle-{X₃₁₀₂,X₃₁₀₃,PU(1)} of X₆

• X₃₀₉₅ = intersection of lines PU(1) of pedal triangles of PU(1)

• X₃₀₉₅ = {X₃₇₁,X₃₇₂}-harmonic conjugate of X₁₆₉₁

Notes

Let U be the pedal triangle of the 1st Brocard point of ABC, and let V be the pedal triangle of the 2nd Brocard point. Let X' = X₃₉-of-U and Y' = X₅₁₂-of-U; let X″ = X₃₉-of-V and Y″ = X₅₁₂-of-V. Then X₃₀₉₅ = X'Y'∩X″Y″. (Randy Hutson, September 5 , 2015)

Let X be a point on the 2nd Brocard circle. The locus of X₄ of triangle XPU(1) as X varies is a circle with center X₃₀₉₅. This circle is the reflection of the 2nd Brocard circle in X₃₉. See also X₉₈₂₁. (Randy Hutson, July 20, 2016)

The locus of the orthocenter in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is a circle with center X₃₀₉₅ and segment X₃₅₅₇X₃₅₅₈ as diameter. The circle also passes through X₄ and its antipode X₁₉₄. (Randy Hutson, August 29, 2018)

Let O_A be the circle centered at the A-vertex of the 5th anti-Brocard triangle and passing through A; define O_B and O_C cyclically. X₃₀₉₅ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 28, 2020)