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traces of the orthic axis

K393 is a nK0+ having a pencil of circular polar conic as in Table 47. These circles are the polar conics of the points on the orthic axis we call the circular line of the cubic. They form a pencil of circles having the parallel at X(1990) to the Euler line as radical axis.

X(1990) lies on the line HK. It is the barycentric product of H and X(30), the infinite point of the Euler line.

The orthic axis meets the sidelines at U, V, W and X(1637) is the isobarycenter of these points (it is also called the tripolar centroid of H). The polar conic of X(1637) is the degenerate circle into the line at infinity and the radical axis.

The polar conic of X(523), the infinite point of the orthic axis, is the circle with center X(1637) passing through the Fermat points and orthogonal to the circumcircle.

For any finite point P on the orthic axis, distinct of X(1637), the polar conic is a proper circle whose center is the inverse of P in the circle above.

K393a

The polar conics of U, V, W are the circles of the pencil which pass through U, V, W. They are tangent to the cubic at U, V, W and the common tangents are parallel to the Euler line. It follows that the polar conic of X(30) in the cubic must decompose into the orthic axis and another line (L). (L) contains X(1637) and also X(1636), X(1651). (L) meets the cubic at another real point having also its tangent parallel to the Euler line.

(L) is the locus of points having a polar conic which is a rectangular hyperbola. All these hyperbolas have parallel asymptotes and meet the radical axis at the same points. They have their centers on the line (L) itself.

K393b

Since the polar conics of U, V, W are circles, it is easy to draw the osculating circles at U, V, W to the cubic. Indeed, it is known (theorem of Moutard) that the curvature is twice that of the corresponding polar conic. Hence, the osculating circle C(U) at U is the circle with diameter U and the center of the polar conic of U.

Note that the radical axis of C(U), C(V), C(W) is the parallel at X(1637) to the Euler line.

See the related K604, K628, CL044 and Q087.