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(2 a^4 - b^2 c^2) x (c^2 y^2 - b^2 z^2) = 0

X(1), X(32), X(76), X(3557), X(3558), X(6177), X(6178), X(6179)

P = 2 a^4 - b^2 c^2 : : = X(6179)

isogonal conjugates X(3557)*, X(3558)* of X(3557), X(3558) : these points are now X(6177), X(6178) in ETC


cevians of P

K704 is the isogonal pK that contains the Pappus points X(3557), X(3558) and their isogonal conjugates X(3557)*, X(3558)*.

These Pappus points X(3557) [resp. X(3558)] are the crosspoints of the real foci F1, F2 [resp. imaginary foci F3, F4] of the Steiner inellipse.

In other words, X(3557) is the pole of the focal axis (A1) in the circum-conic (H1) passing through F1, F2 and likewise for X(3558).

The isogonal conjugates X(3557)*, X(3558)* are the corresponding cevapoints of these same foci. Indeed, the crosspoint and cevapoint of two isogonal conjugate points are themselves two isogonal conjugate points.

Note that X(3557), X(3558) also lie on the cubics K028, K289.

K704 is a member of the pencil of isogonal pKs generated by K020 = pK(X6,X384) and K128 = pK(X6,X385). All these cubics have their pivot on the line X(32), X(76). This pencil also contains :

• pK(X6, X736), a circular cubic passing through X(1), X(32), X(76), X(736), X(737).

• pK(X6, X76), passing through X(1), X(32), X(76), X(1670), X(1671), X(1676), X(1677), X(1759). See K1027 and Q142.

• pK(X6, X3972), passing through X(1), X(32), X(76), X(1340), X(1341). Recall that X(3972) = 2 a^4 + b^2 c^2 : : .