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X(1), X(9), X(40), X(1155), X(3307), X(3308), X(8072), X(8073)

Ia, Ib, Ic : vertices of the excentral triangle

A', B', C' : vertices of the extouch triangle

foci of the Mandart inscribed ellipse

U, V, W : traces of the antiorthic axis

Q1, Q2, Q3 : points of (O) on K343, see below

(contributed by Peter Moses, in relation to Q187)

Q188 has two real asymptotes at X(3307), X(3308) hence parallel to those of the Feuerbach hyperbola. They meet at X(6745) = point of intersection of the line X(1)X(8) and the trilinear polar of X(8).

Q188 has two isotropic asymptotes which meet at X(1155) on the curve.

The osculating circle C(A) at A is centered on the midpoint of A and U, i.e. {b-c,b,-c}. The 3 circles C(A), C(B), C(C) are coaxal (with centers on the Gergonne line {241, 514, 650, etc}) and intersect at X(8072) and X(8073), on the line through X(9), X(40).

The inellipse to the excentral triangle and the triangle formed by the other three meets R1, R2, R3 of Q188 with the Bevan/excentral circle IaIbIc is

b^2*(b - c)^2*c^2*x^2 - 2*a*b*c^2*(a*b + a*c + b*c - c^2)*x*y + a^2*(a - c)^2*c^2*y^2 - 2*a*b^2*c*(a*b - b^2 + a*c + b*c)*x*z + 2*a^2*b*c*(a^2 - a*b - a*c - b*c)*y*z + a^2*(a - b)^2*b^2*z^2 = 0, center X(1376), through X{1018, 35338, 55325}.

The corresponding right-hyperbola passing through R1, R2, R3 and X(1768) is b*(a-b-c)*(b-c)*(a+b-c)*c*(a-b+c)*x^2+4*a*(a-b)*b*(a+b-c)*c^2*x*y-a*(a-c)*(a-b-c)*(a+b-c)*c*(a-b+c)*y^2-4*a*b^2*(a-c)*c*(a-b+c)*x*z-4*a^2*b*(a-b-c)*(b-c)*c*y*z+a*(a-b)*b*(a-b-c)*(a+b-c)*(a-b+c)*z^2 = 0, center X(60782), through X{1, 9, 57, 200, 1750, 1768, 3307, 3308, 30353, 38399}, hence homothetic to the Feuerbach hyperbola.

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These points Q1, Q2, Q3 on (O) and K343, also lie on every cubic of a family of pKs with

• pole Ω on psK(X32 x X284, X284, X6), passing through {6, 31, 55, 1333, 2204, 7118},

• pivot P on the isogonal transform of K1272, namely psK(X284, X333, X4), passing through {4, 9, 21, 57, 58, 63, 284, 573, 1817, 34277, 40575},

• isopivot Q on a circum-cubic passing through {1, 3, 19, 55, 197, 610, 1436, 2217, 7169, 10537} and the vertices of the tangential triangle.

Examples : pK(X31, X573), pK(X55, X9), pK(X1333, X1817).