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X(3), X(4), X(23), X(98), X(1177), X(1503), X(1687), X(1688), X(2353), X(14366), X(53764), X(53765), X(53766), X(53767), X(53768), X(53769), X(53773), X(53774), X(53775)

other points below

Geometric properties :

K1321 is a circular cubic with singular focus F = X(19165), the circumcircle-inverse of X(114).

Its orthic line passes through {5, 182, 206, 1348, 1349, 1503, 1676, 1677, 2909, 2980, etc} hence its homothetic under h(F, 2) is the real asymptote of K1321.

For every point P on K1321, the orthocenter of the pedal triangle of P is a point M on the Brocard axis.

There are four points P (not necessarily all real nor distinct) associated to a same point M on the Brocard axis. In the figure above, three points are represented with their respective pedal triangles, all with orthocenter M.

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Points on K1321

Z1 = X(53765) = -a^8-a^6 b^2+a^4 b^4+a^2 b^6-a^6 c^2+3 a^4 b^2 c^2-a^2 b^4 c^2+b^6 c^2+a^4 c^4-a^2 b^2 c^4-4 b^4 c^4+a^2 c^6+b^2 c^6, SEARCH = 1.793446592024071, on the line {3,98}

Z2 = X(53767) = a^2 (a^12-3 a^10 b^2+2 a^8 b^4+2 a^6 b^6-3 a^4 b^8+a^2 b^10-3 a^10 c^2+a^8 b^2 c^2+2 a^4 b^6 c^2-a^2 b^8 c^2+b^10 c^2+2 a^8 c^4-2 a^4 b^4 c^4+2 a^6 c^6+2 a^4 b^2 c^6-2 b^6 c^6-3 a^4 c^8-a^2 b^2 c^8+a^2 c^10+b^2 c^10), SEARCH =-4.111944251922819, on the line {4,98}

Z3 = X(53768) = (a^2-a b+b^2-c^2) (a^2+a b+b^2-c^2) (a^2-b^2-a c+c^2) (a^2-b^2+a c+c^2) (a^10-2 a^8 b^2+2 a^4 b^6-a^2 b^8-2 a^8 c^2+a^6 b^2 c^2+a^4 b^4 c^2+a^2 b^6 c^2-b^8 c^2+a^4 b^2 c^4-4 a^2 b^4 c^4+b^6 c^4+2 a^4 c^6+a^2 b^2 c^6+b^4 c^6-a^2 c^8-b^2 c^8), SEARCH =-0.4692767728720594, on the lines {23,98}, {1177,1503}, circumcircle-inverse of X(15462)

Z4 = a^2 (a^2+b^2-2 c^2) (a^2-2 b^2+c^2) (a^12-3 a^10 b^2+2 a^8 b^4+2 a^6 b^6-3 a^4 b^8+a^2 b^10-3 a^10 c^2+13 a^8 b^2 c^2-12 a^6 b^4 c^2+2 a^4 b^6 c^2-a^2 b^8 c^2+b^10 c^2+2 a^8 c^4-12 a^6 b^2 c^4+10 a^4 b^4 c^4+2 a^6 c^6+2 a^4 b^2 c^6-2 b^6 c^6-3 a^4 c^8-a^2 b^2 c^8+a^2 c^10+b^2 c^10), SEARCH =3.059033198672515, on the line {23,1177}

Z5 = (a^4+2 a^2 b^2+b^4-4 a^2 c^2-4 b^2 c^2+c^4) (a^4-4 a^2 b^2+b^4+2 a^2 c^2-4 b^2 c^2+c^4) (a^8+2 a^6 b^2-2 a^4 b^4-2 a^2 b^6+b^8+2 a^6 c^2-4 a^4 b^2 c^2+2 a^2 b^4 c^2-2 a^4 c^4+2 a^2 b^2 c^4-2 b^4 c^4-2 a^2 c^6+c^8), SEARCH =-3.892276523838356, on the line {23,2353}

Z6 = X(53766) = a^2 (a^4+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-b^2 c^2+c^4) (a^8-3 a^6 b^2+3 a^4 b^4-a^2 b^6-3 a^6 c^2+a^4 b^2 c^2+a^2 b^4 c^2-b^6 c^2+3 a^4 c^4+a^2 b^2 c^4-a^2 c^6-b^2 c^6), SEARCH =-3.806067219845645, on the line {98,1503}

Z7 = X(53769) = (a^2+b^2-c^2) (a^2-b^2+c^2) (a^16-a^14 b^2-3 a^12 b^4+3 a^10 b^6+3 a^8 b^8-3 a^6 b^10-a^4 b^12+a^2 b^14-a^14 c^2-a^12 b^2 c^2+5 a^10 b^4 c^2-a^8 b^6 c^2-3 a^6 b^8 c^2+a^4 b^10 c^2-a^2 b^12 c^2+b^14 c^2-3 a^12 c^4+5 a^10 b^2 c^4-12 a^8 b^4 c^4+6 a^6 b^6 c^4+a^4 b^8 c^4+5 a^2 b^10 c^4-2 b^12 c^4+3 a^10 c^6-a^8 b^2 c^6+6 a^6 b^4 c^6-2 a^4 b^6 c^6-5 a^2 b^8 c^6-b^10 c^6+3 a^8 c^8-3 a^6 b^2 c^8+a^4 b^4 c^8-5 a^2 b^6 c^8+4 b^8 c^8-3 a^6 c^10+a^4 b^2 c^10+5 a^2 b^4 c^10-b^6 c^10-a^4 c^12-a^2 b^2 c^12-2 b^4 c^12+a^2 c^14+b^2 c^14), SEARCH =3.92169246647351, on the lines {4,1177}, {98,2353}