A'B'C' is the circumsymmedial triangle

A = (14.270629564582983, 5.254379083694099)
B = (8.209916805855812, -0.7836343763112382)
C = (14.997007123681371, -0.6701378827021166)
a = Segment(B, C)
b = Segment(C, A)
c = Segment(A, B)
AB = Line(A, B)
BC = Line(B, C)
CA = Line(C, A)
x15 = TriangleCenter(A, B, C, 15)
x16 = TriangleCenter(A, B, C, 16)
circumcircle = Circle(A, B, C)
x3 = TriangleCenter(A, B, C, 3)
x4 = TriangleCenter(A, B, C, 4)
x574 = TriangleCenter(A, B, C, 574)
k001 = Cubic(A, B, C, 1)
A' = Barycenter({A, B, C}, {(-a^(2)), (2 * b^(2)), (2 * c^(2))})
B' = Barycenter({A, B, C}, {(2 * a^(2)), (-b^(2)), (2 * c^(2))})
C' = Barycenter({A, B, C}, {(2 * a^(2)), (2 * b^(2)), (-c^(2))})
k001CircumSymmedial = Cubic(A', B', C', 1)
f = Line(TriangleCenter[A, B, C, 3], TriangleCenter[A, B, C, 6])
MooreNeelley = TriangleCurve(A, B, C, 0 = ((B * C) * a^(2)) + ((A * C) * b^(2)) + ((A * B) * c^(2)) - ((A + B + C) * ((-(((((B * a^(2)) * c^(2)) * (a^(2) - (5 * b^(2)) + c^(2)))) / (a^(4) - ((4 * a^(2)) * b^(2)) + b^(4) - ((4 * a^(2)) * c^(2)) - ((4 * b^(2)) * c^(2)) + c^(4)))) - ((((A * b^(2)) * c^(2)) * ((-5 * a^(2)) + b^(2) + c^(2)))) / (a^(4) - ((4 * a^(2)) * b^(2)) + b^(4) - ((4 * a^(2)) * c^(2)) - ((4 * b^(2)) * c^(2)) + c^(4)) + ((((C * a^(2)) * b^(2)) * ((-a^(2)) - b^(2) + (5 * c^(2))))) / (a^(4) - ((4 * a^(2)) * b^(2)) + b^(4) - ((4 * a^(2)) * c^(2)) - ((4 * b^(2)) * c^(2)) + c^(4)))))
x6 = TriangleCenter(A, B, C, 6)