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K417

X(1), X(4), X(61), X(62), X(147), X(185), X(194), X(511), X(1046), X(2651)

excenters

extraversions of X(1046), X(2651)

foci of the K-ellipse (inellipse with center K when the triangle ABC is acute angle)

S1, S2 defined at the page K019. See also Table 62.

vertices of the triangle T = Q1Q2Q3 in K755

Other points and a larger figure below

For any pivot P, the polar conic of O in pK(X6, P) is a rectangular hyperbola and O is the only point of the plane having this property.

The locus of P such that these rectangular hyperbolas degenerate into two perpendicular lines is a cubic passing through the in/excenters, X(20), X(185), X(1075). For example, when

  • P = X(1), the lines are the parallels at X(1) to the asymptotes of the circumconic with center X(116),
  • P = X(20), the lines are the line at infinity and the Brocard axis,
  • P = X(185), the lines are the parallels at H to the asymptotes of the Jerabek hyperbola,
  • P = X(1075), the lines are X(4)X(51) and X(520)X(647).

K417 is the locus of centers of the degenerate hyperbolas. It is a member of the pencil generated by the Brocard (third) cubic K019 and the McCay hessian cubic K048 which also contains the strophoid K418.

These hyperbolas form a net of conics which can be generated by any three independent conics. Hence, K417 can be seen as the Jacobian of the net i.e. the locus of the points M such that the polar lines of M in these three conics are concurrent at N which also lies on the cubic. In this case, the polar lines of N in these three conics are concurrent at M.

It is convenient to choose the Jerabek and Stammler hyperbolas which are two members of the net. Indeed, these are the polar conics of O in the Orthocubic K006 and the McCay cubic K003. Let us define the mapping f : M -> M' where M' is the intersection of the polar lines of M in these two hyperbolas.

f is a quadratic involution which leaves invariant all the cubics of the pencil generated by K019 and K048. f can be seen as an isoconjugation in the degenerate triangle formed by the line GX(98) and the parallels at S1, S2 to the Brocard axis. The fixed points of this isoconjugation are O, K, the infinite points of the asymptotes of the Jerabek and Stammler hyperbolas. This is the JS involution described in Table 62.

It follows that if P, Q are two points on K417 and P', Q' their respective JS-images then K417 is the locus of M such that the directed line angles (MP, MQ) and (MQ', MP') are equal (mod. π). For example, with (P, Q) = (X1, X4) we have (P', Q') = (X1046, X185). See Table 62 for other examples.

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K417 is a van Rees focal cubic with singular focus X(147), the anticomplement of the Tarry point X(98). It is the locus of contacts of the circles passing through X(61), X(62) with their tangents drawn from X(147). Recall that X(61), X(62) are the isogonal conjugates of the Napoleon points.

Another generation of K417 derives easily. K417 is the locus of the intersections of the circles centered on the Brocard axis whose radical axis is the perpendicular bisector of X(61)X(62) with the corresponding diameter which passes through X(147).

See also K755, further properties of T, for another description.

K417a

K417 meets the altitude AH again at two points lying on the rectangular hyperbola through the in/excenters and whose center is the second intersection of the circumcircle and the perpendicular at A to the Brocard axis. These are the points with coordinates ±a^2 SA : bc SC : bc SB.

K417 meets the bisectors at A on the line through H and the foot on BC of the trilinear polar of X(2501). This line is perpendicular at X(115) to the line OX(115).

K417 contains the four foci of the inconic with center K. The two real foci are F1, F2 on the figure. The third point on this line is S2, on the Brocard circle and on the line GX(147).