K759


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	X(2), X(3), X(4), X(3431), X(7607), X(9716), X(9717) four foci of the MacBeath inconic : X(3), X(4) and two imaginary Q1, Q2, Q3 : vertices of the Thomson triangle infinite points of pK(X6, X381) isogonal conjugates of the X3-OAP points, see Table 53

(Peter Moses, private message, 2016-03-05)

K759 is spK(X381, X5) as defined in CL055. Its isogonal transform is K762 = spK(X2, X5).

K759 is a member of the net of circum-cubics K(P) = spK(P, Q = midpoint GP) passing through G and the vertices of the Thomson triangle Q1Q2Q3. Furthermore, K(P) contains :

• the infinite points of pK(X6, P),

• the isogonal conjugate P* of P,

• the foci of the inconic with center Q.

• two common points of the line GP and the circum-conic through X6, P*. These two points are the "last" common points of K(P) with the Thomson cubic K002 and also with pK(X6, P).

With P = u : v : w, the equation of K(P) is : ∑ u x [c^2 y (x-2 y+z) - b^2 z (x+y-2 z)] = 0.

Other members of this net are : K002 = K(X2), K280 = K(X6), K581 = K(X3), K615 = K(X20), K1093 = K(X1992).