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X(2), X(3), X(4), X(74), X(110), X(125), X(131), X(541)
foci of the inscribed Steiner ellipse
singular focus F, see below
K816 is a focal cubic with singular focus F = X(14685), the intersection of the lines X(3)X(1495) and X(6)X(647).
K816 is invariant under the Psi transformation which is the product of the reflection about one axis of the Steiner inellipse and the inversion with circle that of diameter F1F2, the foci of the ellipse. See also "Orthocorrespondence and Orthopivotal Cubics", §5 and K018, K022. K816 is a Psi-cubic as in Table 60.
In particular, F = Psi(X74). F lies on the circumcircle of the following triangles : X(2)X(3)X(110), X(2)X(4)X(125), X(3)X(4)X(74), X(74)X(110)X(125) and also on the circle with diameter X(1344)X(1345), a member of the pencil generated by the circumcircle of ABC and its nine points circle.
The orthic line passes through X(2) and X(74) hence the polar conic of any point on this line is a rectangular hyperbola. In particular, the polar conic (H) of X(74) contains the four foci of the inscribed Steiner ellipse and its center is G. The polar conic of X(2) has center X(74) and is homothetic to the Kiepert hyperbola.
• K816 is the locus of contacts of tangents drawn through F to the circles passing through X(2) and X(74).
• K816 is the locus of points from which the segments X(3)X(4) and X(110)X(125) – or equivalently X(3)X(110) and X(4)X(125) – are seen under equal or supplementary angles.
Let P be a finite point not lying on the Euler line, Q its reflection about O, T the midpoint of HP. The centroid of HPQ is then the centroid G of ABC.
The locus of points from which the segments OH and QT – or equivalently OQ and HT – are seen under equal or supplementary angles is a focal cubic K(P) passing through G, O, H, P, Q, T, the infinite point of GP (which is the orthic line of the cubic) and whose singular focus F lies on the circumcircle of triangles X(2)X(3)Q, X(2)X(4)T, X(3)X(4)P, PQT.
When P lies on the circumcircle, F also lies on the circle with diameter X(1344)X(1345) also passing through X(6), X(2453), and on the trilinear polar of P, a line passing through X(6).
The mapping P –> F is quadratic and the locus of P such that the line PF passes through a fixed point M is a focal cubic passing through X(2), X(3), X(4), M. This cubic is a circum-cubic if and only if M = X(6) giving the cubic K072.
The polar conic of P is a rectangular hyperbola with center G and the polar conic of G is a rectangular hyperbola with center P.
The following table gives a selection of these cubics K(P). The points G, O, H are not repeated. The red point is the singular focus.
• P1649 = 2,1499 /\ 69,2418
• P2770 on the line 353,1499
• P9169 = 4,1649 /\ 20,1499