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X(523), X(18015), X(23282), X(23283), X(23284), X(23285), X(23286), X(23287), X(23288), X(23289), X(23290)

centers of the Apollonius circles

Geometric properties :

K1072 = cK(#X523, X6) = nK(X115, X6, X523) is the transform of the Kiepert hyperbola under the mapping 𝛕 defined in page K1065.

Recall that 𝛕 sends any point P in the plane to the intersection P' of the trilinear polar of P and the polar of P in (O). It is given by :

𝛕 : P = u : v : w → P' = u^2 (c^2 v - b^2 w) : v^2 (a^2 w - c^2 u) : w^2 (b^2 u - a^2 v).

The pivotal conic of K1072 is the parabola (P) with focus X(2079), directrix the line (D) = X(5)X(6). (P) is tangent at X(6587) to the orthic axis which is an asymptote of the cubic.

K1072 is a trident since it is tritangent at X(523) to the line at infinity hence the orthic axis is an inflexional asymptote to the cubic. K1072 has also a parabolic asymptote (P1) with equation : ∑ SB SC x^2 + [a^2 SA + (b^2 - c^2)^2] y z = 0 hence X(523) is a sextactic point on K1072.

(P1) meets the orthic axis again at X(6587) and the tangent at this point to (P1) passes through X(2450).

This tangent is perpendicular at X(6587) to (D) and contains the three finite points of inflexion but one only, say F, is real. Its isoconjugate F* is the real common point of (P) and the cubic which are both tangent to the line FF*.

See CL029 for a generalization of cKs which are tridents. See also CL063 for psKs.