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X(99), X(115)

vertices of the cevian triangle of X(115)

P1, P2 on the perpendicular at X(115) to the Euler line and on the circum-conic through G and X(99)

E1, E2 on the circumcircle and on the line through X(115), X(2799).

other points below

K392 = pK(X523, X115) is an example of pK having a pencil of circular polar conics as in Table 47.

It is a member of the class CL009 also containing K601. See the related class CL049 containing the analogous cubics K602, K603. For nK0 with the same property, see CL044, K393, K604.

The polar conic of any point on the line (L) passing through G, X(1637), X(2799) is a circle and all these circles form a pencil. (L) is called the circular line of the cubic.

In particular :

  1. the polar conic of G decomposes into the line at infinity and the radical axis of these circles. It is the perpendicular at X(115) to the Euler line.
  2. the polar conic of X(1637) is the circle with center H.
  3. the polar conic of X(2799) (a point at infinity) is the orthoptic circle of the Steiner in-ellipse.

 
K392X2X523

K392 meets the Steiner ellipse at A, B, C, X(99) and two other points S1, S2 which are the bicentric mates of X(99).

If X(99) = u : v : w then S1 = v : w : u and S2 = w : u : v.

The cevian lines of G and X(523) meet at A, B, C and six other points that lie on K392. With X(523) = U : V : W = 1/u : 1/v : 1/w, these points are :

Ab = U : W : W, Ac = U : V : V,

Bc = U : V : U, Ba = W : V : W,

Ca = V : V : W, Cb = U : U : W.

Furthermore, each cevian line of S1 and S2 contains one of these points. Note that X(115) is the midpoint of S1S2.

These 8 points are two by two collinear with X(115) hence they are isoconjugate points on the cubic.

***
K392X3413X3414

The infinite points of the axes of the Steiner ellipse are X(3413), X(3414). The cevian lines of these points (i.e. the parallels through A, B, C to the axes of the Steiner ellipse) meet at A, B, C and six other points that lie on K392.

These 6 points and P1, P2 described above are two by two collinear with X(115) hence they are isoconjugate points on the cubic.

Furthermore, each cevian line of P1 and P2 contains one of these points. Note that X(115) is the midpoint of P1P2.

 
K392hess

The poloconic (PC) of the line at infinity is the locus of point M whose polar conic (C) is a parabola. Concerning K392, this poloconic is the reunion of the axes of the Steiner ellipse.

(PC) splits the plane into two regions:

• for any M in the yellow region, (C) is an ellipse,

• for any M in the blue region, (C) is a hyperbola.

In particular,

• for any M on the orthic line, (C) is a rectangular hyperbola,

• for any M on the circular line, (C) is a circle.

Since G lies on both lines, its polar conic must decompose into the line at infinity and the radical axis of the circles above.

The hessian (H) is the locus of M whose polar conic decomposes into two lines. (H) contains G, X(523), X(671), X(3413), X(3414). It has three real asymptotes which are the axes of the Steiner ellipse and the perpendicular at X(671) to the Euler line.