X₃₀₉₄

INTERSECTION OF LINES X₂X_694 AND X₃X_6

X₃₀₉₄ = midpoint of X₆₉ and X₁₉₄

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

X₃₀₉₄ (INTERSECTION OF LINES X₂X_694 AND X₃X_6) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {69, 194}

• Related Centers: X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE), X₁₉₄ (-CEVA CONJUGATE OF X₂)

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

Search Value (6, 9, 13): \(2.136050797860\)

▸ Alternate Coordinates2 forms

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

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

▸ Incident Central Lines16 lines

{2, 694}, {3, 6}, {22, 1915}, {69, 194}, {76, 141}, {99, 737}, {262, 1513}, {394, 2056}, {538, 599}, {542, 1569}, {726, 2321}, {1180, 2979}, {1194, 1613}, {1352, 2549}, {1469, 2276}, {2275, 3056}

▸ Functional Relations18 relations

• X₃₀₉₄ = midpoint of X₆₉ and X₁₉₄

• X₃₀₉₄ = reflection of X_i in X_j for these (i,j): (6, 39), (76, 141)

• X₃₀₉₄ = isogonal conjugate of X₃₄₀₇

• X₃₀₉₄ = isotomic conjugate of X₃₁₁₄

• X₃₀₉₄ = Brocard-circle-inverse of X₁₆₉₁

• X₃₀₉₄ = 2nd-Brocard-circle-inverse of X₃₂

• X₃₀₉₄ = crosssum of X₃₂ and X₁₈₂

• X₃₀₉₄ = crosspoint of X₇₆ and X₂₆₂

• X₃₀₉₄ = reflection of X₆ in line PU(1)

• X₃₀₉₄ = X₇₆-of-1st-Brocard-triangle

• X₃₀₉₄ = X₆-of-5th-Brocard-triangle

• X₃₀₉₄ = X₄-of-X₃PU(1)

• X₃₀₉₄ = inverse-in-circle-{{X₃₁₀₂,X₃₁₀₃,PU(1)}} of X₃

• X₃₀₉₄ = perspector of 2nd Brocard triangle and unary cofactor triangle of 1st Brocard triangle

• X₃₀₉₄ = 1st-Brocard-isogonal conjugate of X₄₀₄₈

• X₃₀₉₄ = 1st-Brocard-isotomic conjugate of X₃₇₃₄

• X₃₀₉₄ = {X₃₇₁,X₃₇₂}-harmonic conjugate of X₃₃₉₈

• X₃₀₉₄ = {X₁₃₄₀,X₁₃₄₁}-harmonic conjugate of X₃₉

Notes

Let A₂B₂C₂ be the 2nd Brocard triangle. Let A' = inverse-in-Brocard circle of A, and define B' and C' cyclically. Let A₂' = inverse-in-circumcircle of A₂, and define B₂' and C₂' cyclically. Let A″ = B'B₂'∩C'C₂', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₀₉₄. (Randy Hutson, December 26, 2015)

Let A'B'C' be the 3rd Brocard triangle. Then X₃₀₉₄ is the radical center of the circumcircles of A'BC, B'CA, C'AB. (Randy Hutson, December 26, 2015)

Let D = P(1), E = U(1), F = P(2), G = U(2). Let D' = X₄-of-EFG, E' = X₄-of-DFG, F' = X₄-of-DEG, G' = X₄-of-DEF. Then D’E’F'G' is a cyclic quadrilateral whose circumcenter is X₃₀₉₄. (Randy Hutson, December 26, 2015)