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X(2), X(13), X(14), X(30), X(98), X(125), X(6108), X(6109), X(11656), X(11657), X(11658), X(11659)

(contributed by Peter Moses, 2017-01-10)

K892 is the Psi-image of the focal cubic K876. It is also a focal cubic with singular focus X(125). Its real asymptote is the homothetic of the orthic line X(30)X(115) under h(X125, 2).

K892 meets this real asymptote again at X also on the Fermat axis, SEARCH = 5.11209932384344. X is the tangential of X(125).

K892 also contains :

• P = X(13)X(125} /\ X(14)X(16), SEARCH = -19.7882314033603,

• Q = X(14)X(125} /\ X(13)X(15), SEARCH = 3.61980738798976,

• Y = X(30)X(125} /\ orthic axis, SEARCH = 0.975114478349182.

X, Y, P, Q are now X(11656), X(11657), X(11658), X(11659) in ETC (2017-01-11).

K892 is the locus of contacts of tangents drawn through X(125) to the circles passing through X(6108), X(6109) hence with center on the orthic axis of ABC. It follows that K892 is always an unipartite focal cubic.

In particular, the circle passing through X(125), X(1637) is the polar conic of X(125) and its tangent at X(125) passes through X, its tangents at X(6108), X(6109) pass through X(125).

The circle (C) with center X(1637) passes through the Fermat points X(13), X(14). It is the only one which is orthogonal to the circumcircle of ABC and to the orthoptic circle (S) of the Steiner inellipse (and to any other circle of the well known coaxal system they generate).

The polar conic of X(30) is the rectangular hyperbola passing through X(2), X(30), X(98), X(230), X(523) hence X(2), X(98) are the two real centers of anallagmaty of the cubic. It follows that K892 is tangent at X(2) to the Euler line. Hence, K892 is invariant under two inversions :

• that of center X(2) which swaps X(98) and X(125). In other words, K892 is invariant under the inversion with respect to the circle (S). In particular, the points X(13), X(14) are swapped with X(6108), X(6109) respectively.

• that of center X(98) which swaps X(2) and X(125), also X(13) and X(6109), X(14) and X(6108).

K892 is therefore also invariant under the (commutative) product f of these inversions which is a quadratic mapping with singular points X(125) and the circular points at infinity. This involution is in fact the classical conjugation on a focal cubic : for M on K892, let M' be the third point on X(125)M and then N = f(M) is the third intersection of the cubic with the parallel at M' to the Euler line. For example, f maps the points X(2), X(13), X(30), X(6108), P, X onto the points X(98), X(14), X(125), X(6109), Q, Y respectively.

f can also be construed as follows. K892 has a real prehessian which is a stelloid with radial center X(125). f(M) is the center of the polar conic of M with respect to this prehessian.